Chapter 6

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=2\left(3^{x}\right)-18$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5,8} 12.7$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=4^{x-2}-2$$

Explain how the graph of \(y=-3^{x}+7\) can be obtained from the graph of \(y=3^{x}\).

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=3^{2 x}-9^{x+1}$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=2^{3 x}-8^{x-3}$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=8-4 \log _{5} x$$

Solve \(-3^{x}+7=0\) for \(x,\) expressing \(x\) in terms of a base 3 logarithm.

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=9 \log _{3}(3 x)-18$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=\ln (x+2)$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=\ln (x-1)-\ln (x+1)$$

The following equations are identities because they are true for all real mumbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any neal number. $$\ln |x+\sqrt{x^{2}+3}|+\ln |x-\sqrt{x^{2}+3}|=\ln 3$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=7-5 \log x$$

The following equations are identities because they are true for all real mumbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any neal number. $$\ln \left|x^{2}-\sqrt{x^{4}+1}\right|+\ln \left|x^{2}+\sqrt{x^{4}+1}\right|=0$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=3-2 \log _{4}(x-5)$$

Answer each of the following. Suppose \(f(x)\) is the number of cars that can be built for \(x\) dollars. What does \(f^{-1}(1000)\) represent?

The following equations are identities because they are true for all real mumbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any neal number. $$\frac{1}{3} \ln \left(\frac{x^{2}+1}{5}\right)-\frac{1}{3} \ln \left(\frac{x^{2}+4}{5}\right)=\ln \sqrt[3]{\frac{x^{2}+1}{x^{2}+4}}$$

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$x^{2}=2^{x}$$

The following equations are identities because they are true for all real mumbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any neal number. $$\frac{1}{2} \ln \left(\frac{x^{2}}{7}\right)-\frac{1}{2} \ln \left(\frac{x^{4}+x^{2}}{7}\right)=\ln \sqrt{\frac{1}{x^{2}+1}}$$

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$x^{2}-4=e^{x-4}+4$$

Answer each of the following. If a line has nonzero slope \(a\), what is the slope of its reflection across the line \(y=x ?\)

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$\log x=x^{2}-8 x+14$$

Answer each of the following. $$\text { Find } f^{-1}(f(2)), \text { where } f(2)=3$$

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$\ln x=-\sqrt[3]{x+3}$$

The number of species in a sample is approximated by $$S(n)=a \ln \left(1+\frac{n}{a}\right)$$ where \(n\) is the number of individuals in the sample and \(a\) is a constant that indicates the diversity of species in the community. If \(a=0.36,\) find \(S(n)\) for each value of \(n .\) (Hint: \(S(n)\) must be a whole number.) (a) 100 (b) 200 (c) 150 (d) 10

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$e^{x}=\frac{1}{x+2}$$

When sunlight passes through lake water, its initial intensity \(I_{0}\) decreases to a weaker intensity \(I\) at a depth of \(x\) feet according to the formula $$\ln I-\ln I_{0}=-k x$$ where \(k\) is a positive constant. Solve this equation for \(I .\)

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$3^{-x}=\sqrt{x+5}$$

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=\frac{x-5}{x+3} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

Use any method (analytic or graphical) to solve each equation. $$\log _{2} \sqrt{2 x^{2}}-1=0.5$$

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=\frac{-x}{x-4} ;[-2.6,10.6] \text { by }[-4.1,4.1]$$

Use any method (analytic or graphical) to solve each equation. $$\log x^{2}=(\log x)^{2}$$

Use any method (analytic or graphical) to solve each equation. $$\ln \left(\ln e^{-x}\right)=\ln 3$$

Use any method (analytic or graphical) to solve each equation. $$e^{x+\ln 3}=4 e^{x}$$

Life Span of Robins \(\quad\) Use the equation $$y=\frac{2-\log (100-x)}{0.42}$$ from Example 11 for Exercises 115 and 116. Estimate the number of years elapsed for \(75 \%\) of the robins to die.

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=-x^{2}+4$$

Salinity The salinity of the oceans changes with latitude and depth. In the tropics, the salinity increases on the surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$f(x)=31.5+1.1 \log (x+1)$$ models salinity to depths of 1000 meters at a latitude of 57.5". The variable x is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D .\) Global Physical Climatology, Academic Press.) Approximate analytically, to the nearest meter, the depth where the salinity equals 33.

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=(x-1)^{2}$$

Salinity The salinity of the oceans changes with latitude and depth. In the tropics, the salinity increases on the surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$f(x)=31.5+1.1 \log (x+1)$$ models salinity to depths of 1000 meters at a latitude of 57.5". The variable x is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D .\) Global Physical Climatology, Academic Press.) Estimate the salinity at a depth of 500 meters.

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=x^{4}$$

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=x^{4}+x^{2}-6$$

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=-\sqrt{x^{2}-16}$$

Using the given restrictions on the functions, find a formula for \(f^{-1}\). $$f(x)=-x^{2}+4, \quad x \geq 0$$

Using the given restrictions on the functions, find a formula for \(f^{-1}\). $$f(x)=(x-1)^{2}, \quad x \geq 1$$

Using the given restrictions on the functions, find a formula for \(f^{-1}\). $$f(x)=|x-6|, \quad x \geq 6$$

Using the given restrictions on the functions, find a formula for \(f^{-1}\). $$f(x)=x^{4}, \quad x \geq 0$$