Chapter 6

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=\frac{2}{x+3}$$

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\). Beer, 4.8

If \(f(x)=a^{x}\) and \(f(3)=27,\) find each value. (a) \(f(1)\) (b) \(f(-1)\) (c) \(f(2)\) (d) \(f(0)\)

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=\frac{3}{x-4}$$

Solve each problem. The age in years of a female blue whale is approximated by $$t=-2.57 \ln \left(\frac{87-L}{63}\right).$$ where \(L\) is its length in feet. (a) How old is a female blue whale that measures 80 feet? (b) Estimate the length of a female blue whale that is 4 years old. (c) The equation that defines \(t\) has domain \(24

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\). Drinking water, 6.5

Give an exponential function in the form \(f(x)=a^{x}\) whose graph contains the given point. (a) \((3,8)\) (b) \((-3,64)\)

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=\sqrt{6+x}, x \geq-6$$

Solve each problem. The function $$f(x)=27+1.105 \log (x+1)$$ approximates the barometric pressure in inches of mercury at a distance of \(x\) miles from the eye of a hurricane. (a) Approximate the pressure 9 miles from the eye of the humicane. (b) The ordered pair \((99,29.21)\) belongs to this function. What information does it convey?

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{2}{5}$$

A student told a friend, "You must reject any negative solution of an equation involving logarithms." Is this correct?

Use properties of exponents to write each function in the form \(f(t)=k a^{t},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=2^{3 t+4}$$

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=-\sqrt{x^{2}-16}, x \geq 4$$

During the 100 -meter dash, the elapsed time \(T\) for a sprinter to reach a speed of \(x\) meters per second is given by the following function. $$T(x)=-1.2 \ln \left(1-\frac{x}{11}\right)$$ (a) How much time had elapsed when the sprinter was running 0 meters per second? Interpret your answer. (b) At the end of the race, the sprinter was moving at 10.998 meters per second. What was the sprinter's time for this 100 -meter dash? (c) Find \(T^{-1}(x)\) and interpret its meaning.

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{4} \frac{6}{7}$$

Explain why this logarithmic equation has no solution. $$\ln x-\ln (x+1)=\ln 5$$

Use properties of exponents to write each function in the form \(f(t)=k a^{t},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=3^{2 t+3}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{4 x}{x+1}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{6 x}{y}$$

Use a graphing calculator to find the solution of each equation. Round your result to the nearest thousandth. $$1.5^{\log x}=e^{0.5}$$

Use properties of exponents to write each function in the form \(f(t)=k a^{t},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=\left(\frac{1}{3}\right)^{1-2 t}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{3 x}{5-x}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{4 p}{q}$$

Use a graphing calculator to find the solution of each equation. Round your result to the nearest thousandth. $$1.5^{\ln x}=10^{0.5}$$

The graph of \(y=e^{x-3}\) can be obtained by translating the graph of \(y=e^{x}\) to the right 3 units. Find a constant \(C\) such that the graph of \(y=C e^{x}\) is the same as the graph of \(y=e^{x-3}\). Verify your result by graphing both functions.

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{1-2 x}{3 x}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{5} \frac{5 \sqrt{7}}{3 m}$$

Solve each formula for the indicated variable. $$r=p-k \ln t, \text { for } t$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 20,000\) invested at \(1 \%\) annual interest for 4 years compounded (a) annually; (b) semiannually

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{4-x}{5 x}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{2 \sqrt{3}}{5 p}$$

Solve each formula for the indicated variable. $$p=a+\frac{k}{\ln x}, \text { for } x$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 35,000\) invested at \(1.2 \%\) annual interest for 3 years compounded (a) annually; (b) quarterly

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\sqrt{x^{2}-4}, x \geq 2$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 27,500\) invested at \(0.95 \%\) annual interest for 5 years compounded (a) daily \((n=365) ; \quad\) (b) continuously

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\sqrt{x-8}, x \geq 8$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{6}(7 m+3 q)$$

Solve each formula for the indicated variable. $$A=\frac{P i}{1-(1+i)^{-n}}, \text { for } n$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 15,800\) invested at \(0.6 \%\) annual interest for 6.5 years compounded (a) quarterly; (b) continuously

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=5 x^{3}-7$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{k} \frac{p q^{2}}{m}$$

Solve each formula for the indicated variable. $$A=T_{0}+C e^{-k}, \text { for } k$$

Decide which of the two plans will provide a better yield. (Interest rates stated are annual rates.) Plan A: \(\$ 40,000\) invested for 3 years at \(0.5 \%,\) compounded quarterly Plan B: \(\$ 40,000\) invested for 3 years at \(0.4 \%,\) compounded continuously

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=4-3 x^{3}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{x^{5} y^{3}}{3}$$

Decide which of the two plans will provide a better yield. (Interest rates stated are annual rates.) Plan A: \(\$ 50,000\) invested for 10 years at \(1.75 \%,\) compounded daily \((n=365)\) Plan B: \(\$ 50,000\) invested for 10 years at \(1.7 \%,\) compounded continuously

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{x}{4+3 x}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{m} \sqrt{\frac{r^{3}}{5 z^{5}}}$$

Solve each formula for the indicated variable. $$y=A+B\left(1-e^{-C x}\right), \text { for } x$$

Use the table capabilities of a calculator to work Exercises 75 and 76 . You have the choice of investing \(\$ 1000\) at an annual rate of \(5 \%,\) compounded either annually or monthly. Let \(Y_{1}\) represent the investment compounded annually, and let \(\mathrm{Y}_{2}\) represent the investment compounded monthly. Graph both \(Y_{1}\) and \(Y_{2}\), and observe the slight differences in the curves. Then use a table to compare the graphs numerically. What is the difference between the returns for the investments after 1 year, 2 years, 5 years, 10 years, 20 years, 30 years, and 40 years?