Chapter 2

If a function \(f\) is given by the formula \(y=f(x),\) then \(f(a)\) is the ___________ of \(f\) at \(x=a\).

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.

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

For a function \(f,\) the set of all possible inputs is called the ________ of \(f,\) and the set of all possible outputs is called the _______ of \(f\).

By definition, \(f \circ g(x)=\) _____ So if \(g(2)=5\) and f(5)=12, \text { 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)?

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.

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

(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 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 \(x\)

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 _____.

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

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

A function is given algebraically by the formula \(f(x)=\) \((x-4)^{2}+3 .\) Complete these other ways to represent \(f:\) (a) Verbal: "Subtract \(4,\) then __________ and ________ (b) Numerical: $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 19 \\ 2 & \\ 4 & \\ 6 & \\ \hline \end{array}$$

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 \(f^{-1}(f(x))=x\)

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 \((b, f(b))\)

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

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

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

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

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

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

Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=f(x+7)\) (b) \(y=f(x)+7\)

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

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

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

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

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

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 \(\frac{1}{3}\)

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

Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=-2 f(x)\) (b) \(y=-\frac{1}{2} f(x)\)

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

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

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

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

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

A 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. $$f(x)=x-1$$

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

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

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

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

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

A 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. $$f(x)=2(x+1)$$

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

Express the function (or rule) in words. $$f(x)=\frac{x-4}{3}$$

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

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

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

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