Chapter 5

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=-2 x^{2}+x $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\frac{x^{2}-3 x+2}{x+1}, \quad g(x)=\frac{x^{2}-1}{x-2} $$

Simplify the expression. $$ \log _{0,4}\left(\frac{2}{5}\right)^{-3} $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=4-\frac{3}{4} x $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\frac{4 x-2}{x+2}, \quad g(x)=\frac{2 x-1}{3 x+6} $$

Give an example of data that could be modeled by a logistic function and explain why.

Simplify the expression. $$ \ln e^{-4} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. \(f(0)=5\) and for each unit increase in \(x,\) the output is multiplied by 1.5

Use a calculator to approximate each pair of expressions. Then state which property of logarithms this calculation illustrates. $$ \log _{2} \frac{\sqrt{x}}{z^{2}} $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{4} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\frac{2}{x^{2}-1}, \quad g(x)=\frac{x+1}{x^{2}-2 x+1} $$

How can you distinguish data that illustrate exponential growth from data that illustrate logarithmic growth?

Simplify the expression. $$ 2^{\log _{2} k} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. \(f(1)=3\) and for each unit increase in \(x,\) the output is multiplied by \(\frac{3}{4}\)

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=|2 x-5| $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=\frac{1}{x+2}, \quad g(x)=x^{2}+x-2 $$

Simplify the expression. $$ \log _{5} 5^{x} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(0)=10 \text { and } f(1)=20 $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=|x-1| $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x^{5 / 2}-x^{3 / 2}, \quad g(x)=x^{1 / 2} $$

Simplify the expression. $$ \log _{6} 6^{9} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(0)=7 \text { and } f(-1)=1 $$

Use a calculator to approximate each pair of expressions. Then state which property of logarithms this calculation illustrates. $$ \log \frac{\sqrt{x^{2}+4}}{\sqrt[3]{x-1}} $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{3} $$

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x^{2 / 3}-2 x^{1 / 3}+1, \quad g(x)=x^{1 / 3}-1 $$

Simplify the expression. $$ 3^{\log _{3}(x-1)} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(1)=9 \text { and } f(2)=27 $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=\frac{1}{1+x^{2}} $$

Simplify the expression. $$ 8^{\log _{5}(x+1)} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(-1)=\frac{1}{4} \text { and } f(1)=4 $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=\frac{1}{x} $$

Simplify the expression. $$ \log _{2} 64 $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(-2)=\frac{9}{2} \text { and } f(2)=\frac{1}{18} $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=3 x-x^{3} $$

Simplify the expression. $$ \log _{2} \frac{1}{4} $$

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(-2)=\frac{3}{4} \text { and } f(2)=12 $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{2 / 3} $$

Simplify the expression. $$\log _{4} 2$$

Find \(C\) and a so that \(f(x)=C a^{x}\) models the situation described. State what the variable \(x\) represents in your formula. (Answers may vary.) There are initially 5000 bacteria, and this sample doubles in size every hour.

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{1 / 2} $$

Exercises 35 and 36: Use the tables to evaluate each expression, if possible. (a) \((f+g)(-1)\) (b) \((g-f)(0)\) (c)\((g f)(2)\) (d)\((f / g)(2)\) $$ \begin{array}{rrrr} x & -1 & 0 & 2 \\ f(x) & -3 & 5 & 1 \end{array} $$ $$ \begin{array}{rrrr} x & -1 & 0 & 2 \\ g(x) & -2 & 3 & 0 \end{array} $$

Simplify the expression. $$\log _{3} 9$$

Find \(C\) and a so that \(f(x)=C a^{x}\) models the situation described. State what the variable \(x\) represents in your formula. (Answers may vary.) Fifteen hundred dollars is deposited in an account that triples in value every decade.

(Refer to Examples 5 and \(6 .\) ) Write the expression as a logarithm of a single expression. $$ \ln 33-\ln 11 $$

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{3}-4 x $$

Simplify the expression. $$\ln e^{-3}$$

Find \(C\) and a so that \(f(x)=C a^{x}\) models the situation described. State what the variable \(x\) represents in your formula. (Answers may vary.) In 2000 a house was worth \(\$ 200,000,\) and its value decreases by \(5 \%\) each year thereafter.

Exercises 37 and 38: Use the table to evaluate each expression, if possible. (a) \((f+g)(2)\) (b) \((f-g)(4)\) (c) \((f g)(-2)\) (d) \((f / g)(0)\) $$ \begin{array}{rrrrr} x & -2 & 0 & 2 & 4 \\ f(x) & 0 & 5 & 7 & 10 \\ g(x) & 6 & 0 & -2 & 5 \end{array} $$

Modeling Decide if the situation could be modeled by a one-to-one function. The distance between the ground and a person who is riding a Ferris wheel after \(x\) seconds

Simplify the expression. $$\ln e$$