Chapter 3

A function \(f\) is one-to-one if different inputs produce _____ outputs. You can tell from the graph that a function is one-to-one by using the _____ Test.

If you travel 100 miles in two hours, then your average speed for the trip is average speed \(=\frac{\square}{\square}=\)_____.

\(1-2\) Fill in the blank with the appropriate direction (left, right.up, or down). (a) The graph of \(y=f(x)+3\) is obtained from the graph of \(y=f(x)\) by shifting ________3 units. (b) The graph of \(y=f(x+3)\) is obtained from the graph of \(y=f(x)\) by shifting ________3 units.

By definition, \(f \circ g(x)=______\quad\) So if \(g(2)=5\) and \(f(5)=12,\) then \(f \circ g(2)= _______\)

(a) For a function to have an inverse, it must be _____. So which one of the following functions has an inverse? $$ f(x)=x^{2} \quad g(x)=x^{3} $$ (b) What is the inverse of the function that you chose in part (a)?

\(1-2\) Fill in the blank with the appropriate direction (left, right.up, or down). (a) The graph of \(y=f(x)-3\) is obtained from the graph of \(y=f(x)\) by shifting_________3 units. (b) The graph of \(y=f(x-3)\) is obtained from the graph of \(y=f(x)\) by shifting_________3 units.

For a function \(f,\) the set of all possible inputs is called the $$ \begin{array}{l}{\text { of } f, \text { and the set of all possible outputs is called the }} \\ {\text { of } f}\end{array} $$

If \(f(2)=3,\) then the point (2,________) is on the graph of \(f\).

If the rule of the function \(f\) is "add one" and the rule of the function \(g\) is "multiply by 2 " then the rule of \(f \circ g\) is " ___________________" and the rule of \(g \circ f\) is " _______________________"

A function \(f\) has the following verbal description: "Multiply by \(3,\) add \(5,\) and then take the third power of the result." (a) Write a verbal description for \(f^{-1} .\) (b) Find algebraic formulas that express \(f\) and \(f^{-1}\) in terms of the input \(x .\)

The average rate of change of the function \(f(x)=x^{2}\) between \(x=1\) and \(x=5\) is average rate of change \(=\frac{\square}{\square}=\)_____.

Fill in the blank with the appropriate axis (x-axis or \(y\) -axis) (a) The graph of \(y=-f(x)\) is obtained from the graph of \(y=f(x)\) by reflecting in the_________ (b) The graph of \(y=f(-x)\) is obtained from the graph of \(y=f(x)\) by reflecting in the_________

(a) Which of the following functions have 5 in their domain? \(f(x)=x^{2}-3 x \quad g(x)=\frac{x-5}{x} \quad h(x)=\sqrt{x-10}\) (b) For the functions from part (a) that do have 5 in their domain, find the value of the function at \(5 .\)

If the point \((2,3)\) is on the graph of \(f,\) then \(f(2)=\) ________.

True or false? (a) If \(f\) has an inverse, then \(f^{-1}(x)\) is the same as \(\frac{1}{f(x)}\) (b) If \(f\) has an inverse, then \(f^{-1}(f(x))=x\)

(a) The average rate of change of a function \(f\) between \(x=a\) and \(x=b\) is the slope of the _____ line between \((a, f(a))\) and \((b, f(b))\). (b) The average rate of change of the linear function \(f(x)=3 x+5\) between any two points is _____.

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=x-3, \quad g(x)=x^{2} $$

Express the rule in function notation. (For example, the rule "square, then subtract 5\("\) is expressed as the function \(f(x)=x^{2}-5 .\) Add \(3,\) then multiply by 2

Sketch the graph of the function by first making a table of values. \(f(x)=2\)

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=x^{2}+2 x, g(x)=3 x^{2}-1 $$

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) \(\begin{array}{ll}{\text { (a) } y=f(x+7)} & {\text { (b) } y=f(x)+7}\end{array}\)

Express the rule in function notation. (For example, the rule "square, then subtract 5\("\) is expressed as the function \(f(x)=x^{2}-5 .\) Divide by \(7,\) then subtract 4

Sketch the graph of the function by first making a table of values. \(f(x)=-3\)

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=\sqrt{4-x^{2}}, \quad g(x)=\sqrt{1+x} $$

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=-f(x) \quad\) (b) \(y=f(-x)\)

Express the rule in function notation. (For example, the rule "square, then subtract 5\("\) is expressed as the function \(f(x)=x^{2}-5 .\) Subtract \(5,\) then square

Sketch the graph of the function by first making a table of values. \(f(x)=2 x-4\)

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=\sqrt{9-x^{2}}, \quad g(x)=\sqrt{x^{2}-4} $$

Express the rule in function notation. (For example, the rule "square, then subtract 5\("\) is expressed as the function \(f(x)=x^{2}-5 .\) Take the square root, add \(8,\) then multiply by \(\frac{1}{3}\)

Sketch the graph of the function by first making a table of values. \(f(x)=6-3 x\)

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=\frac{2}{x}, g(x)=\frac{4}{x+4} $$

A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=3 x-2 ; \quad x=2, x=3 $$

\(9-18\) m function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .(b)\) Find the domain and range of \(f\) from the graph. $$ f(x)=x-1 $$

Express the function (or rule) in words. $$ h(x)=x^{2}+2 $$

Sketch the graph of the function by first making a table of values. \(f(x)=-x+3, \quad-3 \leq x \leq 3\)

Find \(f+g, f-g \cdot f g,\) and \(f / g\) and their domains. $$ f(x)=\frac{2}{x+1}, \quad g(x)=\frac{x}{x+1} $$

A function is given. Determine the average rate of change of the function between the given values of the variable. $$ g(x)=5+\frac{1}{2} x, \quad x=1, x=5 $$

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=f(x-4)+\frac{3}{4} \quad\) (b) \(y=f(x+4)-\frac{3}{4}\)

\(9-18\) m function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .(b)\) Find the domain and range of \(f\) from the graph. $$ f(x)=2(x+1) $$

Express the function (or rule) in words. $$ k(x)=\sqrt{x+2} $$

Sketch the graph of the function by first making a table of values. \(f(x)=\frac{x-3}{2}, \quad 0 \leq x \leq 5\)

Find the domain of the function. $$ f(x)=\sqrt{x}+\sqrt{1-x} $$

Determine whether the function is one-to-one. $$ f(x)=-2 x+4 $$

A function is given. Determine the average rate of change of the function between the given values of the variable. $$ h(t)=t^{2}+2 t, \quad t=-1, t=4 $$

\(5-14\) . Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f .\) (a) \(y=2 f(x+1)-3 \quad\) (b) \(y=2 f(x-1)+3\)

Sketch the graph of the function by first making a table of values. \(f(x)=-x^{2}\)

\(9-18\) m function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .(b)\) Find the domain and range of \(f\) from the graph. $$ f(x)=4, \quad 1 \leq x \leq 3 $$

Find the domain of the function. $$ g(x)=\sqrt{x+1}-\frac{1}{x} $$

Determine whether the function is one-to-one. $$ f(x)=3 x-2 $$

A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(z)=1-3 z^{2} ; \quad z=-2, z=0 $$