Review of Functions: Fresh Take

  • Identify zeros of functions using equations, graphs, and tables
  • Interpret graphs and tables to describe their behavior including properties of symmetry
  • Make new functions from two or more given functions

Functions

The Main Idea 

  • A function is a special type of relation where each input is associated with exactly one output.
  • Functions are used to describe relationships between two sets.
  • The input of a function is called the independent variable, often denoted as [latex]x[/latex].
  • The output of a function is called the dependent variable, often denoted as [latex]y[/latex].
  • Function notation: [latex]y = f(x)[/latex] is read as “[latex]y[/latex] equals [latex]f[/latex] of [latex]x[/latex].”
  • Functions can be represented using tables, graphs, or formulas (algebraic expressions).

Consider the function [latex]g(x) = x² - 4x + 3[/latex]

  1. Evaluate [latex]g(3)[/latex]
  2. Create a small table of values for [latex]x = 0, 1, 2, 3, 4[/latex]

The Domain and Range of a Function

The Main Idea 

  • The domain of a function is the complete set of possible input values ([latex]x[/latex]-values).
  • The range of a function is the complete set of possible output values ([latex]y[/latex]-values).
  • Domain restrictions can occur due to mathematical limitations (e.g., division by zero, square roots of negative numbers).
  • Set notation and interval notation are used to describe domains and ranges.
  • The union symbol ([latex]\cup[/latex]) is used to describe domains and ranges with gaps or breaks.

Determine the domain and range of the function[latex]f(x) = |x + 1| - 3[/latex].

Intercepts of a Function

The Main Idea 

  • [latex]x[/latex]-intercepts are points where a function’s graph crosses the [latex]x[/latex]-axis ([latex]f(x) = 0[/latex]).
  • [latex]x[/latex]-intercepts are also called zeros or roots of the function.
  • The [latex]y[/latex]-intercept is the point where a function’s graph crosses the [latex]y[/latex]-axis ([latex]x = 0[/latex]).
  • A function can have multiple [latex]x[/latex]-intercepts but at most one y-intercept.
  • [latex]x[/latex]-intercepts provide information about the shape of the function’s graph.
  • The [latex]y[/latex]-intercept represents the function’s output when the input is zero.

Consider the function [latex]f(x)=\sqrt{x+3}+1[/latex].

  1. Find all zeros of [latex]f[/latex].
  2. Find the [latex]y[/latex]-intercept (if any).
  3. Sketch a graph of [latex]f[/latex].

Find the zeros of [latex]f(x)=x^3-5x^2+6x[/latex].

Symmetry of Functions

The Main Idea 

  • Function Symmetry:
    • Symmetry about the [latex]y[/latex]-axis: [latex]f(x) = f(-x)[/latex] (even functions)
    • Symmetry about the origin: [latex]f(-x) = -f(x)[/latex] (odd functions)
    • Some functions are neither even nor odd
  • Absolute Value Function:
    • Defined as [latex]f(x) = |x| = x[/latex]if [latex]x ≥ 0[/latex], and [latex]-x[/latex] if [latex]x < 0[/latex]
    • Always outputs non-negative values
    • Symmetric about the [latex]y[/latex]-axis (even function)

An image of two graphs. The first graph is labeled “(a), symmetry about the y-axis” and is of the curved function “f(x) = (x to the 4th) - 2(x squared) - 3”. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. This function decreases until it hits the point (-1, -4), which is minimum of the function. Then the graph increases to the point (0,3), which is a local maximum. Then the the graph decreases until it hits the point (1, -4), before it increases again. The second graph is labeled “(b), symmetry about the origin” and is of the curved function “f(x) = x cubed - 4x”. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. The graph of the function starts at the x intercept at (-2, 0) and increases until the approximate point of (-1.2, 3.1). The function then decreases, passing through the origin, until it hits the approximate point of (1.2, -3.1). The function then begins to increase again and has another x intercept at (2, 0).

Figure 13. (a) A graph that is symmetric about the [latex]y[/latex]-axis. (b) A graph that is symmetric about the origin.

If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[/latex] has symmetry? Looking at Figure 13(a) again, we see that since [latex]f[/latex] is symmetric about the [latex]y[/latex]-axis, if the point [latex](x,y)[/latex] is on the graph, the point [latex](−x,y)[/latex] is on the graph. In other words, [latex]f(−x)=f(x)[/latex]. If a function [latex]f[/latex] has this property, we say [latex]f[/latex] is an even function, which has symmetry about the [latex]y[/latex]-axis. For example, [latex]f(x)=x^2[/latex] is even because

[latex]f(−x)=(−x)^2=x^2=f(x)[/latex].

In contrast, looking at Figure 13(b) again, if a function [latex]f[/latex] is symmetric about the origin, then whenever the point [latex](x,y)[/latex] is on the graph, the point [latex](−x,−y)[/latex] is also on the graph. In other words, [latex]f(−x)=−f(x)[/latex]. If [latex]f[/latex] has this property, we say [latex]f[/latex] is an odd function, which has symmetry about the origin. For example, [latex]f(x)=x^3[/latex] is odd because

[latex]f(−x)=(−x)^3=−x^3=−f(x)[/latex].

Determine whether [latex]f(x)=4x^3-5x[/latex] is even, odd, or neither.

Absolute Value Functions

For the function [latex]f(x)=|x+2|-4[/latex], find the domain and range.

Composing Functions

The Main Idea 

  1. Combining Functions with Mathematical Operators:
    • Sum: [latex](f + g)(x) = f(x) + g(x)[/latex]
    • Difference :[latex](f - g)(x) = f(x) - g(x)[/latex]
    • Product: [latex](f · g)(x) = f(x) · g(x)[/latex]
    • Quotient: [latex](\frac{f}{g})(x) = \frac{f(x)}{g(x)}\text{, where }g(x) ≠ 0[/latex]
  2. Composite Functions:
    • [latex](f ∘ g)(x) = f(g(x))[/latex]
    • The output of g becomes the input of f
    • Order matters: [latex](f ∘ g)(x) ≠ (g ∘ f)(x)[/latex] in general

In general [latex]f\circ g[/latex] and [latex]g\circ f[/latex] are different functions. In other words in many cases [latex]f\left(g\left(x\right)\right)\ne g\left(f\left(x\right)\right)[/latex] for all [latex]x[/latex].

For example if [latex]f\left(x\right)={x}^{2}[/latex] and [latex]g\left(x\right)=x+2[/latex], then

[latex]\begin{align}f\left(g\left(x\right)\right)&=f\left(x+2\right) \\[2mm] &={\left(x+2\right)}^{2} \\[2mm] &={x}^{2}+4x+4\hfill \end{align}[/latex]

but

[latex]\begin{align}g\left(f\left(x\right)\right)&=g\left({x}^{2}\right) \\[2mm] \text{ }&={x}^{2}+2\hfill \end{align}[/latex]

Consider the functions [latex]f(x)=x^2+1[/latex] and [latex]g(x)=\frac{1}{x}[/latex].

  1. Find [latex](g\circ f)(x)[/latex] and state its domain and range.
  2. Evaluate [latex](g\circ f)(4)[/latex] and [latex](g\circ f)\left(-\frac{1}{2}\right)[/latex].

Let [latex]f(x)=2-5x[/latex]

Let [latex]g(x)=\sqrt{x}[/latex]

Find [latex](f\circ g)(x)[/latex]

Consider the functions [latex]f[/latex] and [latex]g[/latex] described below.

[latex]x[/latex] [latex]-3[/latex] [latex]-2[/latex] [latex]-1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex]
[latex]f(x)[/latex] [latex]0[/latex] [latex]4[/latex] [latex]2[/latex] [latex]4[/latex] [latex]-2[/latex] [latex]0[/latex] [latex]-2[/latex] [latex]4[/latex]

 

[latex]x[/latex] [latex]-4[/latex] [latex]-2[/latex] [latex]0[/latex] [latex]2[/latex] [latex]4[/latex]
[latex]g(x)[/latex] [latex]1[/latex] [latex]0[/latex] [latex]3[/latex] [latex]0[/latex] [latex]5[/latex]
  1. Evaluate [latex](f\circ f)(3)[/latex] and [latex](f\circ f)(1)[/latex].
  2. State the domain and range of [latex](f\circ f)(x)[/latex].

If items are on sale for [latex]10\%[/latex] off their original price, and a customer has a coupon for an additional [latex]30\%[/latex] off, what will be the final price for an item that is originally [latex]x[/latex] dollars, after applying the coupon to the sale price?