Chapter 4

True or False? Suppose \(f\) is a one-to-one function with domain all real numbers. Then there is only one solution to the equation \(f(x)=4\)

Complete them to review topics relevant to the remaining exercises. An exponential function of the form \(f(x)=C a^{x},\) where \(C>0\) and \(a>1,\) models exponential __________.

In Exercises \(1-4,\) rewrite using rational exponents. $$\sqrt[5]{x}$$

$$\text {Rewrite using rational exponents.}$$ $$\sqrt{3}$$

Evaluate the expression. $$5^{3}$$

These exercises correspond to the Just in Time references in this section. Complete them to review topics relevant to the remaining exercises. The composite function \(f \circ g\) is defined as \((f \circ g)(x)=\) _______. (a) \(f(g(x))\) (b) \(g(f(x))\)

True or False? \(f(x)=2 x+3\) is not a one-to-one function.

In Exercises \(1-4,\) rewrite using rational exponents. $$\sqrt[3]{z}$$

$$\text {Rewrite using rational exponents.}$$ $$\sqrt{5}$$

Evaluate the expression. $$8^{1 / 3}$$

Find \((f \circ g)(x)\) $$f(x)=x+2, g(x)=x^{2}-2$$

True or False? \(f(x)=e^{x}\) is not a one-to-one function.

Complete them to review topics relevant to the remaining exercises. Let \(f(x)=5 e^{x} .\) As \(x \rightarrow \infty, f(x) \rightarrow\)__________.

In Exercises \(1-4,\) rewrite using rational exponents. $$\sqrt[5]{x^{3}}$$

$$\text {Rewrite using rational exponents.}$$ $$\sqrt[3]{10}$$

Evaluate the expression. $$2^{-2}$$

Find \((f \circ g)(x)\) $$f(x)=\frac{1}{x}, g(x)=\frac{1}{x+3}$$

True or False? \(f(x)=\ln x\) is a one-to-one function.

Complete them to review topics relevant to the remaining exercises. Let \(f(x)=5 e^{x} .\) As \(x \rightarrow-\infty, f(x) \rightarrow\)__________.

In Exercises \(1-4,\) rewrite using rational exponents. $$\sqrt[3]{y^{2}}$$

$$\text {Rewrite using rational exponents.}$$ $$\sqrt[3]{12}$$

Evaluate the expression. $$3^{1 / 2}$$

Find \((f \circ g)(x)\) $$f(x)=\sqrt{x}, g(x)=(x+4)^{2}$$

Solve the exponential equation. Round to three decimal places, when needed. $$5^{x}=125$$

Complete them to review topics relevant to the remaining exercises. Let \(f(x)=\left(\frac{1}{3}\right)^{x} .\) As \(x \rightarrow \infty, f(x) \rightarrow\)__________.

True or False? \(x^{-1}=\frac{1}{x}\)

\(f\) and \(g\) are inverses of each other. True or False? \((f \circ g)(x)=x\)

Evaluate the expression. $$2\left(3^{2}\right)$$

Simplify the expression. $$\sqrt[3]{x^{3}}$$

Solve the exponential equation. Round to three decimal places, when needed. $$7^{2 x}=49$$

Complete them to review topics relevant to the remaining exercises. Let \(f(x)=\left(\frac{1}{2}\right)^{x} .\) As \(x \rightarrow-\infty, f(x) \rightarrow\)__________.

True or False? \(\frac{y}{x^{3}}=x^{3} y^{-1}\)

\(f\) and \(g\) are inverses of each other. True or False? The domain of \(f\) equals the domain of \(g\).

Evaluate the expression. $$2^{3} 2^{5}$$

Simplify the expression. $$\sqrt[3]{7 x^{3}}$$

Solve the exponential equation. Round to three decimal places, when needed. $$10^{x}=1000$$

Use \(f(t)=10 e^{-t}\). Evaluate \(f(0)\).

In Exercises \(7-14,\) use \(\log 2 \approx 0.3010, \log 5 \approx 0.6990,\) and \(\log 7 \approx 0.8451\) to evaluate each logarithm without using a calculator. Then check your answer using a calculator. $$\log 35$$

\(f\) and \(g\) are inverses of each other. True or False? The domain of \(f\) equals the range of \(g\).

Evaluate each expression to four decimal places using a calculator. $$2.1^{1 / 3}$$

Simplify the expression. $$\sqrt[3]{4\left(\frac{1}{4} x^{3}\right)}$$

Solve the exponential equation. Round to three decimal places, when needed. $$10^{x}=0.0001$$

Use \(f(t)=10 e^{-t}\). Evaluate \(f(2)\).

\(f\) and \(g\) are inverses of each other. True or False? \(f\) is a one-to-one function.

Evaluate each expression to four decimal places using a calculator. $$3.2^{1 / 2}$$

Simplify the expression. $$\sqrt[3]{2\left(\frac{1}{2} x^{3}+\frac{5}{2}\right)-5}$$

Solve the exponential equation. Round to three decimal places, when needed. $$4^{x}=\frac{1}{16}$$

Use \(f(t)=10 e^{-t}\). For what value of \(t\) will \(f(t)=5 ?\)

In Exercises \(7-14,\) use \(\log 2 \approx 0.3010, \log 5 \approx 0.6990,\) and \(\log 7 \approx 0.8451\) to evaluate each logarithm without using a calculator. Then check your answer using a calculator. $$\log \frac{2}{5}$$

Write 8,450,000 in scientific notation.