{"id":1641,"date":"2025-07-25T16:50:11","date_gmt":"2025-07-25T16:50:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1641"},"modified":"2025-12-17T15:09:40","modified_gmt":"2025-12-17T15:09:40","slug":"introduction-to-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-functions-get-stronger\/","title":{"raw":"Introduction to Functions: Get Stronger","rendered":"Introduction to Functions: Get Stronger"},"content":{"raw":"

Functions and Function Notation<\/h2>\r\n1. What is the difference between a relation and a function?\r\n\r\n3. Why does the vertical line test tell us whether the graph of a relation represents a function?\r\n\r\n5. Why does the horizontal line test tell us whether the graph of a function is one-to-one?\r\n\r\nFor the following exercises, determine whether the relation represents a function.\r\n\r\n7. [latex]{(a,b),(b,c),(c,c)}[\/latex]\r\n\r\nFor the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].\r\n\r\n10. [latex]x=y^{2}[\/latex]\r\n\r\n[reveal-answer q=\"Q10\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q10\"]\r\nTry solving the equation for `y`. Ask yourself: can one value of `x` produce more than one value of `y`?\r\n\r\n[\/hidden-answer]\r\n\r\n19. [latex]2xy=1[\/latex]\r\n\r\n25. [latex]y^2=x^2[\/latex]\r\n\r\n[reveal-answer q=\"Q25\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q25\"]\r\nIf you take a square root, remember that both the positive and negative roots are possible.\r\n\r\n[\/hidden-answer]\r\n\r\nFor the following exercises, evaluate the function [latex]f[\/latex] at the indicated values [latex]f(\u22123),f(2),f(\u2212a),\u2212f(a),f(a+h)[\/latex].\r\n\r\n27. [latex]f(x)=2x\u22125[\/latex]\r\n\r\n33. Given the function [latex]g(x)=x^{2}+2x[\/latex],evaluate [latex]\\frac{g(x)\u2212g(a)}{x\u2212a},x\\ne{a}[\/latex].\r\n\r\n[reveal-answer q=\"Q33\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q33\"]\r\nExpand both expressions fully, then look for a common factor in the numerator that matches the denominator.\r\n\r\n[\/hidden-answer]\r\n\r\nFor the following exercises, use the vertical line test to determine which graphs show relations that are functions.For the following exercises, use the function [latex]f[\/latex] represented in the table below.\r\n
\r\n
<\/div>\r\n<\/section><\/div>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/td>\r\n[latex]f<\/span>(<\/span>x<\/span>)[\/latex]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/td>\r\n<\/tr>\r\n
0<\/td>\r\n74<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n28<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n53<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n56<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n36<\/td>\r\n<\/tr>\r\n
7<\/td>\r\n45<\/td>\r\n<\/tr>\r\n
8<\/td>\r\n14<\/td>\r\n<\/tr>\r\n
9<\/td>\r\n47<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n66. Evaluate [latex]f(3)[\/latex].\r\n\r\n67. Solve\u00a0[latex]f(x)=1[\/latex].\r\n\r\n89. The number of cubic yards of dirt,\u00a0[latex]D[\/latex],needed to cover a garden with area a square feet is given by\u00a0[latex]D=g(a)[\/latex].\r\nA garden with area 5000 ft2<\/sup> requires 50 yd<sup>3<\/sup> of dirt. Express this information in terms of the function [latex]g[\/latex].\r\nExplain the meaning of the statement\u00a0[latex]g(100)=1[\/latex].\r\n\r\n91. 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:\r\n[latex]h(1)=200[\/latex]\r\n[latex]h(2)=350[\/latex]\r\n

Domain and Range<\/h2>\r\n
\r\n\r\n1. Why does the domain differ for different functions?\r\n\r\n3. Explain why the domain of [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex] is different from the domain of [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex].\r\n\r\n4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?\r\n

For the following exercises, find the domain of each function using interval notation.<\/p>\r\n8. [latex]f\\left(x\\right)=3\\sqrt{x - 2}[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=3-\\sqrt{6 - 2x}[\/latex]\r\n\r\n14. [latex]f\\left(x\\right)=\\frac{9}{x - 6}[\/latex]\r\n\r\n16. [latex]f\\left(x\\right)=\\frac{\\sqrt{x+4}}{x - 4} [\/latex]\r\n\r\n[reveal-answer q=\"Q16\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q16\"]\r\nCheck *both* conditions: the expression inside the square root must be nonnegative, and the denominator cannot be zero.\r\n\r\n[\/hidden-answer]\r\n\r\n20. [latex]\\frac{5}{\\sqrt{x - 3}} [\/latex]\r\n\r\n[reveal-answer q=\"Q20\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q20\"]\r\nBecause the square root is in the denominator, the expression inside the radical must be *strictly greater than zero*.\r\n\r\n[\/hidden-answer]\r\n\r\n22. [latex]f\\left(x\\right)=\\frac{\\sqrt{x - 4}}{\\sqrt{x - 6}} [\/latex]\r\n\r\n[reveal-answer q=\"Q22\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q22\"]\r\nAnalyze each square root separately. Remember that a square root in the denominator cannot equal zero.\r\n\r\n[\/hidden-answer]\r\n\r\nFor the following exercises, write the domain and range of each function using interval notation.\r\n\r\n27.\r\n\r\n\"Graph\r\n\r\n29.\r\n\r\n\"Graph\r\n\r\n<\/div>\r\n

\r\n\r\n33.\r\n\r\n\"Graph\r\n\r\n[reveal-answer q=\"Q33_DR\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q33_DR\"]\r\nAn unbounded graph in one direction means the domain or range extends to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\nFor the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.\r\n\r\n39. [latex]f\\left(x\\right)=\\begin{cases}{2x - 1}&\\text{ if }&{ x }<{ 1 }\\\\ {1+x }&\\text{ if }&{ x }\\ge{ 1 } \\end{cases}[\/latex]\r\n\r\n[reveal-answer q=\"Q38\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q38\"]\r\nGraph each rule only on the interval where it applies. Use open or closed circles to show whether endpoints are included.\r\n\r\n[\/hidden-answer]\r\n\r\n41. [latex]f\\left(x\\right)=\\begin{cases}{3} &\\text{ if }&{ x } <{ 0 }\\\\ \\sqrt{x}&\\text{ if }&{ x }\\ge { 0 }\\end{cases}[\/latex]\r\n\r\n[reveal-answer q=\"Q41\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q41\"]\r\nTreat each rule separately, then combine them into one graph. Pay attention to what happens at the boundary value.\r\n\r\n[\/hidden-answer]\r\n\r\n47. [latex]f\\left(x\\right)=\\begin{cases}{ 1 }&\\text{ if }&{ x }\\le{ -3 }\\\\{ 0 }&\\text{ if }&{ x }>{ -3 }\\end{cases}[\/latex]\r\n\r\n49. [latex]f\\left(x\\right)=\\begin{cases}{ 7x+3 }&\\text{ if }&{ x }<{ 0 }\\\\{ 7x+6 }&\\text{ if }&{ x }\\ge{ 0 }\\end{cases}[\/latex]\r\n

Rates of Change and Behaviors of Graphs<\/h2>\r\n2. If a function [latex]f[\/latex] is increasing on [latex]\\left(a,b\\right)[\/latex] and decreasing on [latex]\\left(b,c\\right)[\/latex], then what can be said about the local extremum of [latex]f[\/latex] on [latex]\\left(a,c\\right)?[\/latex]\r\n\r\n3. How are the absolute maximum and minimum similar to and different from the local extrema?\r\n\r\nFor exercises 5\u201315, find the average rate of change of each function on the interval specified for real numbers [latex]b[\/latex] or [latex]h[\/latex].\r\n\r\n5. [latex]f\\left(x\\right)=4{x}^{2}-7[\/latex] on [latex]\\left[1,\\text{ }b\\right][\/latex]\r\n\r\n[reveal-answer q=\"Q5_ARC\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q5_ARC\"]\r\nUse the average rate of change formula: change in output divided by change in input.\r\n\r\n[\/hidden-answer]\r\n\r\n7. [latex]p\\left(x\\right)=3x+4[\/latex] on [latex]\\left[2,\\text{ }2+h\\right][\/latex]\r\n\r\n[reveal-answer q=\"Q7_ARC\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q7_ARC\"]\r\nFor a linear function, the average rate of change is the same as the slope.\r\n\r\n[\/hidden-answer]\r\n\r\n15. [latex]\\frac{f\\left(x+h\\right)-f\\left(x\\right)}{h}[\/latex] given [latex]f\\left(x\\right)=2{x}^{2}-3x[\/latex] on [latex]\\left[x,x+h\\right][\/latex]\r\n\r\n[reveal-answer q=\"Q15\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q15\"]\r\nExpand `f(x+h)` first, then factor out `h` so it cancels with the denominator.\r\n\r\n[\/hidden-answer]\r\n\r\nFor exercises 16\u201317, consider the graph of [latex]f[\/latex].\"Graph\r\n\r\n16. Estimate the average rate of change from [latex]x=1[\/latex] to [latex]x=4[\/latex].\r\n\r\n[reveal-answer q=\"Q16_G\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q16_G\"]\r\nEstimate the change in y-values over the change in x-values between the two points.\r\n\r\n[\/hidden-answer]\r\n\r\n17. Estimate the average rate of change from [latex]x=2[\/latex] to [latex]x=5[\/latex].\r\n\r\nFor exercises 22\u201323, consider the graph shown\u00a0below.\r\n\"Graph\r\n\r\n22. Estimate the intervals where the function is increasing or decreasing.\r\n\r\n[reveal-answer q=\"Q22_G\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q22_G\"]\r\nIncreasing and decreasing intervals always use parentheses and only uses the x-coordinates.\r\n[\/hidden-answer]\r\n\r\n23. Estimate the point(s) at which the graph of [latex]f[\/latex] has a local maximum or a local minimum.\r\n\r\n44.\u00a0At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?\r\n\r\n[reveal-answer q=\"Q44\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"Q44\"]\r\nAverage speed equals total distance traveled divided by total time.\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"

Functions and Function Notation<\/h2>\n

1. What is the difference between a relation and a function?<\/p>\n

3. Why does the vertical line test tell us whether the graph of a relation represents a function?<\/p>\n

5. Why does the horizontal line test tell us whether the graph of a function is one-to-one?<\/p>\n

For the following exercises, determine whether the relation represents a function.<\/p>\n

7. [latex]{(a,b),(b,c),(c,c)}[\/latex]<\/p>\n

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

10. [latex]x=y^{2}[\/latex]<\/p>\n