Chapter 1

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$(f \circ g)^{-1}$$

Does every line have both an \(x\) -intercept and a \(y\) -intercept? Explain.

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$10+3 x$$

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$(g \circ f)^{-1}$$

Can every line be written in slope-intercept form? Explain.

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$\frac{x}{3}-5 x^{2}+x^{3}$$

The table shows men's shoe sizes in the United States and the corresponding European shoe sizes. Let \(y=f(x)\) represent the function that gives the men's European shoe size in terms of \(x,\) the men's U.S. size. $$\begin{array}{|c|c|}\hline \text { Men’s U.S. } & \text { Men’s European } \\\\\text { shoe size } & \text { shoe size } \\\\\hline 8 & 41 \\\9 & 42 \\\10 & 43 \\\11 & 44 \\\12 & 45 \\\13 & 46 \\\\\hline\end{array}$$ (a) Is \(f\) one-to-one? Explain. (b) Find \(f(11)\). (c) Find \(f^{-1}(43),\) if possible. (d) Find \(f\left(f^{-1}(41)\right)\). (e) Find \(f^{-1}(f(12))\).

Does every line have an infinite number of lines that are parallel to it? Explain.

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$7 x^{4}+\sqrt{2} x^{2}-x$$

Let \(y=g(x)\) represent the function that gives the women's European shoe size in terms of \(x,\) the women's U.S. size. A women's U.S. size 6 shoe corresponds to a European size \(37 .\) Find \(g^{-1}(g(6)).\)

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=-x^{2}-x+3\) (a) \(f(4)\) (b) \(f(-5)\) (c) \(f(x-2)\)

You can encode and decode messages using functions and their inverses. To code a message, first translate the letters to numbers using 1 for "A," 2 for "B," and so on. Use 0 for a space. So, "A ball" becomes 1 0 2 1 12 12. Then, use a one-to-one function to convert to coded numbers. Using \(f(x)=2 x-1,\) "A ball" becomes 1 ?1 3 1 23 23. (a) Encode "Call me later" using the function \(f(x)=5 x+4.\) (b) Find the inverse function of \(f(x)=5 x+4\) and use it to decode 119 44 9 104 4 104 49 69 29.

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$x+20$$

Evaluate the function at each specified value of the independent variable and simplify. \(f(x)=x \sqrt{x-3}\) (a) \(f(3)\) (b) \(f(12)\) (c) \(f(6)\)

Your wage is \(\$ 12.00\) per hour plus \(\$ 0.55\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced \(x\) is \(y=12+0.55 x.\) (a) Find the inverse function. What does each variable in the inverse function represent? (b) Use a graphing utility to graph the function and its inverse function. (c) Use the trace feature of the graphing utility to find the hourly wage when 9 units are produced per hour. (d) Use the trace feature of the graphing utility to find the number of units produced per hour when your hourly wage is \(\$ 21.35\)

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$3 x-10 x^{2}+1$$

Find the difference quotient and simplify your answer. $$f(x)=x^{2}-2 x+9, \frac{f(3+h)-f(3)}{h}, h \neq 0$$

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$4 x^{2}+x^{-1}-3$$

Find the difference quotient and simplify your answer. $$f(x)=5+6 x-x^{2}, \frac{f(6+h)-f(6)}{h}, h \neq 0$$

If the inverse function of \(f\) exists, and the graph of \(f\) has a \(y\)-intercept, then the \(y\)-intercept of \(f\) is an \(x\)-intercept of \(f^{-1}.\)

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$2 x^{2}-2 x^{4}-x^{3}+\sqrt{2}$$

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$\frac{x^{2}+3 x+4}{x^{2}-9}$$

Factor the trinomial. $$x^{2}-6 x-27$$

Factor the trinomial. $$x^{2}+11 x+28$$

Writing Describe the relationship between the graph of a function \(f\) and the graph of its inverse function \(f^{-1}\).

Factor the trinomial. $$2 x^{2}+11 x-40$$

Think About It The domain of a one-to-one function \(f\) is [0,9] and the range is \([-3,3] .\) Find the domain and range of \(f^{-1}.\)

Factor the trinomial. $$3 x^{2}-16 x+5$$

Think About It The function \(f(x)=\frac{9}{5} x+32\) can be used to convert a temperature of \(x\) degrees Celsius to its corresponding temperature in degrees Fahrenheit. (a) Using the expression for \(f,\) make a conceptual argument to show that \(f\) has an inverse function. (b) What does \(f^{-1}(50)\) represent?

Think About It A function \(f\) is increasing over its entire domain. Does \(f\) have an inverse function? Explain.

Think About It Describe a type of function that is not one-to-one on any interval of its domain.

Proof Prove that if \(f\) and \(g\) are one-to-one functions, then \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\).

Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

Write the rational expression in simplest form. $$\frac{27 x^{3}}{3 x^{2}}$$

Write the rational expression in simplest form. $$\frac{5 x^{2} y^{2}+25 x^{2} y}{x y+5 x}$$

Write the rational expression in simplest form. $$\frac{x^{2}-36}{6-x}$$

Write the rational expression in simplest form. $$\frac{x^{2}+3 x-40}{x^{2}-3 x-10}$$

Determine whether the equation represents \(y\) as a function of \(x .\) $$x=5$$

Determine whether the equation represents \(y\) as a function of \(x .\) $$y-7=-3$$

Determine whether the equation represents \(y\) as a function of \(x .\) $$x^{2}+y=5$$

Determine whether the equation represents \(y\) as a function of \(x .\) $$x-y^{2}=0$$