Functions: Get Stronger

Functions and Function Notation

Verbal

  1. What is the difference between a relation and a function?
  2. Why does the vertical line test tell us whether the graph of a relation represents a function?
  3. Why does the horizontal line test tell us whether the graph of a function is one-to-one?

Algebraic

For the following exercises, determine whether the relation represents [latex]y[/latex] as a function of [latex]x[/latex].

  1. [latex]y=x^2[/latex]
  2. [latex]3x^2+y=14[/latex]
  3. [latex]y=−2x^2+40x[/latex]
  4. [latex]x=\frac{3y+5}{7y−1}[/latex]

For the following exercises, evaluate [latex]f(−3),f(2),f(−a),−f(a),f(a+h)[/latex].

  1. [latex]f(x)=2x−5[/latex]
  2. [latex]f(x)=\sqrt{2−x}+5[/latex]
  3. [latex]f(x)=|x−1|−|x+1|[/latex]

Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

  1. Graph of relation.
  2. Graph of relation.
  3. Graph of relation.

For the following exercises, determine if the given graph is a one-to-one function.

  1. Graph of a circle.
  2. Graph of a rotated cubic function.
  3. Graph of a one-to-one function.

Numeric

For the following exercises, determine whether the relation represents a function.

  1. [latex]{(3,4),(4,5),(5,6)}[/latex]

For the following exercises, determine whether the relation represents [latex]y[/latex] as a function of [latex]x[/latex].

  1. [latex]x[/latex] 5 10 15
    [latex]y[/latex] 3 8 14
  2. [latex]x[/latex] 5 10 10
    [latex]y[/latex] 3 8 14

For the following exercise, use the function [latex]f[/latex] represented in the table below.

  1. Solve [latex]f(x)=1[/latex].
    [latex]x[/latex] 0 1 2 3 4 5 6 7 8 9
    [latex]f(x)[/latex] 74 28 1 53 56 3 36 45 14 47

For the following exercises, evaluate the function [latex]f[/latex] at the values [latex]f(−2),f(−1),f(0),f(1),[/latex] and [latex]f(2)[/latex].

  1. [latex]f(x)=f(x)=8−3x[/latex]
  2. [latex]f(x)=3+\sqrt{x+3}[/latex]
  3. [latex]f(x)=3^x[/latex]

For the following exercise, evaluate the expression, given functions [latex]f[/latex],[latex]g[/latex], and [latex]h[/latex]:

[latex]f(x)=3x−2[/latex]

[latex]g(x)=5−x^2[/latex]

[latex]h(x)=−2x^2+3x−1[/latex]

  1. [latex]f(\frac{7}{3})−h(−2)[/latex]

For the following exercise, graph [latex]y=x^2[/latex] on the given domain. Determine the corresponding range. Show each graph.

  1. [latex][−10,10][/latex]

For the following exercises, graph [latex]y=x^3[/latex] on the given domain. Determine the corresponding range. Show each graph.

  1. [latex][−0.1,0.1][/latex]
  2. [latex][−100,100][/latex]

For the following exercise, graph [latex]y=\sqrt{x}[/latex] on the given domain. Determine the corresponding range. Show each graph.

  1. [latex][0,100][/latex]

For the following exercises, graph [latex]y=\sqrt[3]{x}[/latex] on the given domain. Determine the corresponding range. Show each graph.

  1. [latex][−0.001, 0.001][/latex]
  2. [latex][−1,000,000, 1,000,000][/latex]

Real-World Applications

  1. The number of cubic yards of dirt, [latex]D[/latex], needed to cover a garden with area [latex]a[/latex] square feet is given by [latex]D=g(a)[/latex].
    1. A garden with area [latex]5000 ft^2[/latex] requires [latex]50 yd^3[/latex] of dirt. Express this information in terms of the function [latex]g[/latex].
    2. Explain the meaning of the statement [latex]g(100)=1[/latex].
  2. Let [latex]h(t)[/latex] be the height above ground, in feet, of a rocket [latex]t[/latex] seconds after launching. Explain the meaning of each statement:
    1. [latex]h(1)=200[/latex]
    2. [latex]h(2)=350[/latex]

Linear Functions

Verbal

  1. Terry is skiing down a steep hill. Terry’s elevation, [latex]E(t)[/latex], in feet after t seconds is given by [latex]E(t)=3000−70t[/latex]. Write a complete sentence describing Terry’s starting elevation and how it is changing over time.
  2. A boat is [latex]100[/latex] miles away from the marina, sailing directly toward it at [latex]10[/latex] miles per hour. Write an equation for the distance of the boat from the marina after [latex]t[/latex]hours.

Algebraic

For the following exercises, determine whether the equation of the curve can be written as a linear function.

  1. [latex]y=3x-5[/latex]
  2. [latex]3x+5y=15[/latex]
  3. [latex]3x+5y^2=15[/latex]

For the following exercises, determine whether each function is increasing or decreasing.

  1. [latex]g(x)=5x+6[/latex]
  2. [latex]b(x)=8−3x[/latex]
  3. [latex]k(x)=−4x+1[/latex]

For the following exercises, find the slope of the line that passes through the two given points.

  1. [latex](1,5)[/latex] and [latex](4,11)[/latex]
  2. [latex](8,–2)[/latex] and [latex](4,6)[/latex]

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.

  1. [latex]f(−5)=−4[/latex] and [latex]f(5)=2[/latex]
  2. Passes through [latex](2,4)[/latex] and [latex](4,10)[/latex]
  3. Passes through [latex](-1,4)[/latex] and [latex](5,2)[/latex]

For the following exercises, find the x– and y-intercepts of each equation.

  1. [latex]f(x)=−x+2[/latex]
  2. [latex]h(x)=3x−5[/latex]
  3. [latex]−2x+5y=20[/latex]

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. 

  1. Line 1: Passes through [latex](0,6)[/latex] and [latex](3,−24)[/latex]
    Line 2: Passes through [latex](−1,19)[/latex] and [latex](8,−71)[/latex]
  2. Line 1: Passes through [latex](2,3)[/latex] and [latex](4,−1)[/latex]
    Line 2: Passes through [latex](6,3)[/latex] and [latex](8,5)[/latex]
  3. Line 1: Passes through [latex](2,5)[/latex] and [latex](5,−1)[/latex]
    Line 2: Passes through [latex](−3,7)[/latex] and [latex](3,−5)[/latex]

For the following exercises, write an equation for the line described.

  1. Write an equation for a line parallel to [latex]g(x)=3x−1[/latex] and passing through the point [latex](4,9)[/latex].
  2. Write an equation for a line perpendicular to [latex]p(t)=3t+4[/latex] and passing through the point [latex](3,1)[/latex].

Graphical

For the following exercise, find the slope of the line graphed.

  1. This is a graph of a function on an x, y coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The lines passes through points at (0, -2) and (2, -2).

For the following exercises, write an equation for the line graphed.

  1. Graph of a decreasing linear function with points (0,5) and (4,0)
  2. Graph of an increasing linear function with points at (1,2) and (0,-2)
  3. Graph of a function with points at (0,-2.5) and (-2.5,-2.5)

For the following exercises, match the given linear equation with its graph in the figure below.

Graph of six functions where line A has a slope of 3 and y-intercept at 2, line B has a slope of 1 and y-intercept at 2, line C has a slope of 0 and y-intercept at 2, line D has a slope of -1/2 and y-intercept at -1, line E has a slope of -1 and y-intercept at -1, and line F has a slope of -2 and y-intercept at -1.
  1. [latex]f(x)=−3x−1[/latex]
  2. [latex]f(x)=2[/latex]
  3. [latex]f(x)=3x+2[/latex]

For the following exercises, sketch a line with the given features.

  1. An x-intercept [latex](–2,0)[/latex] and y-intercept of [latex](0,4)[/latex]
  2. A y-intercept of [latex](0,3)[/latex] and slope [latex]\frac{2}{5}[/latex]
  3. Passing through the points [latex](–3,–4)[/latex] and [latex](3,0)[/latex]

For the following exercises, sketch the graph of each equation.

  1. [latex]f(x)=−3x+2[/latex]
  2. [latex]f(x)=\frac{2}{3}x−3[/latex]

For the following exercises, write the equation of the line shown in the graph.

  1. The graph of a line with a slope of 0 and y-intercept at 3.
  2. Graph of a line with an undefined slope and x-intercept at -3.

Numeric

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

  1. [latex]x[/latex] [latex]0[/latex] [latex]5[/latex] [latex]10[/latex] [latex]15[/latex]
    [latex]g(x)[/latex] [latex]5[/latex] [latex]-10[/latex] [latex]-25[/latex] [latex]-40[/latex]
  2. [latex]x[/latex] [latex]0[/latex] [latex]5[/latex] [latex]10[/latex] [latex]15[/latex]
    [latex]f(x)[/latex] [latex]-5[/latex] [latex]20[/latex] [latex]45[/latex] [latex]70[/latex]

Technology

For the following exercises, use a calculator or graphing technology to complete the task.

  1. If [latex]f[/latex] is a linear function, [latex]f(0.1)=11.5[/latex], and [latex]f(0.4)=–5.9[/latex], find an equation for the function.
  2. Graph the function [latex]f[/latex] on a domain of [latex][–10,10]: f(x)=2,500x+4,000[/latex].
  3. The table shows the input, [latex]p[/latex], and output, [latex]q[/latex], for a linear function [latex]q[/latex].
    1. Fill in the missing values of the table.
    2. Write the linear function [latex]q[/latex].
[latex]p[/latex] [latex]0.5[/latex] [latex]0.8[/latex] [latex]12[/latex] [latex]b[/latex]
[latex]q[/latex] [latex]400[/latex] [latex]700[/latex] [latex]a[/latex] [latex]1,000,000[/latex]
  1. Graph the linear function [latex]f[/latex] on a domain of [latex][−0.1,0.1][/latex] for the function whose slope is [latex]75[/latex] and y-intercept is [latex]−22.5[/latex]. Label the points for the input values of [latex]−0.1[/latex] and [latex]0.1[/latex].

Real-World Applications

  1. A gym membership with two personal training sessions costs [latex]$125[/latex], while gym membership with five personal training sessions costs [latex]$260[/latex]. What is cost per session?
  2. A phone company charges for service according to the formula: [latex]C(n)=24+0.1n[/latex], where [latex]n[/latex] is the number of minutes talked, and [latex]C(n)[/latex] is the monthly charge, in dollars. Find and interpret the rate of change and initial value.
  3. A city’s population in the year 1960 was [latex]287,500[/latex]. In 1989 the population was [latex]275,900[/latex]. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.
  4. Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: [latex]I(x)=1054x+23,286[/latex], where [latex]x[/latex] is the number of years after 1990. Which of the following interprets the slope in the context of the problem?
    1. As of 1990, average annual income was [latex]$23,286[/latex].
    2. In the ten-year period from 1990–1999, average annual income increased by a total of [latex]$1,054[/latex].
    3. Each year in the decade of the 1990s, average annual income increased by [latex]$1,054[/latex].
    4. Average annual income rose to a level of [latex]$23,286[/latex] by the end of 1999.

Quadratic Functions

Verbal

  1. Explain the advantage of writing a quadratic function in standard form.
  2. Explain why the condition of [latex]a≠0[/latex] is imposed in the definition of the quadratic function.
  3. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

  1. [latex]g(x)=x^2+2x−3[/latex]
  2. [latex]f(x)=x^2+5x−2[/latex]
  3. [latex]k(x)=3x^2−6x−9[/latex]

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

  1. [latex]f(x)=2x^2−10x+4[/latex]
  2. [latex]f(x)=4x^2+x−1[/latex]
  3. [latex]f(x)=\frac{1}{2}x^2+3x+1[/latex]

For the following exercises, determine the domain and range of the quadratic function.

  1. [latex]f(x)=(x−3)^2+2[/latex]
  2. [latex]f(x)=x^2+6x+4[/latex]
  3. [latex]k(x)=3x^2−6x−9[/latex]

For the following exercises, use the vertex [latex](h,k)[/latex] and a point on the graph [latex](x,y)[/latex] to find the general form of the equation of the quadratic function.

  1. [latex](h,k)=(−2,−1),(x,y)=(−4,3)[/latex]
  2. [latex](h,k)=(2,3),(x,y)=(5,12)[/latex]
  3. [latex](h,k)=(3,2),(x,y)=(10,1)[/latex]

Graphical

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

  1. [latex]f(x)=x^2−6x−1[/latex]
  2. [latex]f(x)=x^2−7x+3[/latex]
  3. [latex]f(x)=4x^2−12x−3[/latex]

For the following exercises, write the equation for the graphed quadratic function.

  1. Graph of a positive parabola with a vertex at (-1, 2) and y-intercept at (0, 3)
  2. Graph of a negative parabola with a vertex at (-1, 2).

Numeric

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

  1. [latex]x[/latex] [latex]-2[/latex] [latex]-1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex]
    [latex]y[/latex] [latex]1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]4[/latex] [latex]9[/latex]
  2. [latex]x[/latex] [latex]-2[/latex] [latex]-1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex]
    [latex]y[/latex] [latex]-8[/latex] [latex]-3[/latex] [latex]0[/latex] [latex]1[/latex] [latex]0[/latex]

Power Functions and Polynomial Functions

Verbal

  1. Explain the difference between the coefficient of a power function and its degree.
  2. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.
  3. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As [latex]x→−∞[/latex], [latex]f(x)→−∞[/latex] and as [latex]x→∞[/latex], [latex]f(x)→−∞[/latex].

Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

  1. [latex]f(x)=(x^2)^3[/latex]
  2. [latex]f(x)=\frac{x^2}{x^2−1}[/latex]
  3. [latex]f(x)=3^{x+1}[/latex]

For the following exercises, find the degree and leading coefficient for the given polynomial.

  1. [latex]7−2x^2[/latex]
  2. [latex]x(4−x^2)(2x+1)[/latex]

For the following exercises, determine the end behavior of the functions.

  1. [latex]f(x)=x^4[/latex]
  2. [latex]f(x)=−x^4[/latex]
  3. [latex]f(x)=−2x^4−3x^2+x−1[/latex]

For the following exercises, find the intercepts of the functions.

  1. [latex]f(t)=2(t−1)(t+2)(t−3)[/latex]
  2. [latex]f(x)=x^4−16[/latex]
  3. [latex]f(x)=x(x^2−2x−8)[/latex]

Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

  1. Graph of an odd-degree polynomial.
  2. Graph of an odd-degree polynomial.
  3. Graph of an odd-degree polynomial.

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

  1. Graph of an odd-degree polynomial.
  2. Graph of an even-degree polynomial.

Numeric

For the following exercises, make a table to confirm the end behavior of the function.

  1. [latex]f(x)=x^4−5x^2[/latex]
  2. [latex]f(x)=(x−1)(x−2)(3−x)[/latex]

Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

  1. [latex]f(x)=x^3(x−2)[/latex]
  2. [latex]f(x)=x(14−2x)(10−2x)[/latex]
  3. [latex]f(x)=x^3−16x[/latex]

Graphs of Polynomial Functions

Verbal

  1. What is the difference between an [latex]x[/latex]– intercept and a zero of a polynomial function [latex]f[/latex]?
  2. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.
  3. If the graph of a polynomial just touches the [latex]x[/latex]-axis and then changes direction, what can we conclude about the factored form of the polynomial?

Algebraic

For the following exercises, find the [latex]x[/latex]– or [latex]t[/latex]-intercepts of the polynomial functions.

  1. [latex]C(t)=3(t+2)(t−3)(t+5)[/latex]
  2. [latex]C(t)=2t(t−3)(t+1)^2[/latex]
  3. [latex]C(t)=4t^4+12t^3−40t^2[/latex]
  4. [latex]f(x)=x^3+x^2−20x[/latex]

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

  1. [latex]f(x)=x^3−9x[/latex], between [latex]x=2[/latex] and [latex]x=4[/latex].
  2. [latex]f(x)=−x^4+4[/latex], between [latex]x=1[/latex] and [latex]x=3[/latex].
  3. [latex]f(x)=x^3−100x+2[/latex], between [latex]x=0.01[/latex] and [latex]x=0.1[/latex].

For the following exercises, find the zeros and give the multiplicity of each.

  1. [latex]f(x)=x^2(2x+3)^5(x−4)^2[/latex]
  2. [latex]f(x)=x^2(x^2+4x+4)[/latex]
  3. [latex]f(x)=(3x+2)^5(x^2−10x+25)[/latex]

Graphical

For the following exercises, graph the polynomial functions. Note [latex]x[/latex]– and [latex]y[/latex]– intercepts, multiplicity, and end behavior.

  1. [latex]g(x)=(x+4)(x−1)^2[/latex]
  2. [latex]k(x)=(x−3)^3(x−2)^2[/latex]
  3. [latex]n(x)=−3x(x+2)(x−4)[/latex]

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

  1. Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.
  2. Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.

For the following exercises, use the graph to identify zeros and multiplicity.

  1. Graph of a negative even-degree polynomial with zeros at x=-4, -2, 1, and 3.
  2. Graph of a positive even-degree polynomial with zeros at x=-2,, and 3.

For the following exercises, use the given information about the polynomial graph to write the equation.

  1. Degree [latex]3[/latex]. Zeros at [latex]x=–2, x=1[/latex], and [latex]x=3[/latex]. [latex]y[/latex]-intercept at [latex](0,–4)[/latex].
  2. Degree [latex]5[/latex]. Roots of multiplicity [latex]2[/latex] at [latex]x=3[/latex] and [latex]x=1[/latex], and a root of multiplicity [latex]1[/latex] at [latex]x=–3[/latex]. [latex]y[/latex]-intercept at [latex](0,9)[/latex].
  3. Degree [latex]5[/latex]. Double zero at [latex]x=1[/latex], and triple zero at [latex]x=3[/latex]. Passes through the point [latex](2,15)[/latex].

Technology

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

  1. [latex]f(x)=x^3−x−1[/latex]
  2. [latex]f(x)=x^4+x[/latex]
  3. [latex]f(x)=x^4−x^3+1[/latex]

Dividing Polynomials

Algebraic

For the following exercises, use long division to divide. Specify the quotient and the remainder.

  1. [latex](x^2+5x−1)÷(x−1)[/latex]
  2. [latex](3x^2+23x+14)÷(x+7)[/latex]
  3. [latex](6x^2−25x−25)÷(6x+5)[/latex]

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)

  1. [latex](2x^3−6x^2−7x+6)÷(x−4)[/latex]
  2. [latex](4x^3−12x^2−5x−1)÷(2x+1)[/latex]
  3. [latex](3x^3−2x^2+x−4)÷(x+3)[/latex]
  4. [latex](2x^3+7x^2−13x−3)÷(2x−3)[/latex]

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

  1. [latex]x−2[/latex], [latex]3x^4−6x^3−5x+10[/latex]
  2. [latex]x−2[/latex], [latex]4x^4−15x^2−4[/latex]
  3. [latex]x+\frac{1}{3}[/latex], [latex]3x^4+x^3−3x+1[/latex]

Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

  1. Factor is [latex](x^2+2x+4)[/latex]
    Graph of a polynomial that has a x-intercept at 1.
  2. Factor is [latex]x^2+x+1[/latex]
    Graph of a polynomial that has a x-intercept at 5.

For the following exercises, use synthetic division to find the quotient and remainder.

  1. [latex]\frac{4x^3−33}{x−2}[/latex]
  2. [latex]\frac{3x^3+2x−5}{x−1}[/latex]
  3. [latex]\frac{x^4−22}{x+2}[/latex]

Zeros of Polynomial Functions

Verbal

  1. Describe a use for the Remainder Theorem.
  2. What is the difference between rational and real zeros?
  3. If synthetic division reveals a zero, why should we try that value again as a possible solution?

Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.

  1. [latex](3x^3−2x^2+x−4)÷(x+3)[/latex]
  2. [latex](−3x^2+6x+24)÷(x−4)[/latex]
  3. [latex](x^4−1)÷(x−4)[/latex]
  4. [latex](4x^3+5x^2−2x+7)÷(x+2)[/latex]

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

  1. [latex]f(x)=2x^3+x^2−5x+2; x+2[/latex]
  2. [latex]f(x)=2x^3+3x^2+x+6; x+2[/latex]
  3. [latex]x^3+3x^2+4x+12; x+3[/latex]
  4. [latex]2x^3+5x^2−12x−30; 2x+5[/latex]

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.

  1. [latex]2x^3+7x^2−10x−24=0[/latex]
  2. [latex]x^3+5x^2−16x−80=0[/latex]
  3. [latex]2x^3−3x^2−32x−15=0[/latex]
  4. [latex]2x^3−3x^2−x+1=0[/latex]

Numeric

For the following exercises, list all possible rational zeros for the functions.

  1. [latex]f(x)=2x^3+3x^2−8x+5[/latex]
  2. [latex]f(x)=6x^4−10x^2+13x+1[/latex]

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

  1. [latex]f(x)=6x^3−7x^2+1[/latex]
  2. [latex]f(x)=8x^3−6x^2−23x+6[/latex]