Chapter 6

(a) Explain why a polynomial function of even degree with domain \((-\infty, \infty)\) cannot be one-to-one. (b) Explain why in some cases a polynomial function of odd degree with domain \((-\infty, \infty)\) is not one-to-one.

Solve each equation. Give the exact answer. $$\log _{x} 81=4$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$1.2(0.9)^{x}=0.6$$

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=e^{x}-1$$

Solve each equation. Give the exact answer. $$\log _{2}(x+1)=3$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$3(2)^{x-2}+1=100$$

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=10^{x}$$

Answer each of the following. For \(f\) to be one-to-one, if \(a \neq b,\) then__________.

Solve each equation. Give the exact answer. $$\log _{3}(x-1)=2$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$5(1.2)^{3 x-2}+1=11$$

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=10^{-x}$$

Answer each of the following. If \(f\) and \(g\) are inverses, then \((f \circ g)(x)=\) __________ and __________$$=x$$.

Solve each equation. Give the exact answer. $$\log _{9} \frac{\sqrt[4]{27}}{3}=x$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$2(1.05)^{x}+3=10$$

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=4^{-x}$$

Answer each of the following. The domain of \(f\) is equal to the __________ of \(f^{-1},\) and the range of \(f\) is equal to the __________ $$\text { of } f^{-1}$$.

Solve each equation. Give the exact answer. $$\log _{1 / 4} \frac{16^{2}}{2^{-3}}=x$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$3(1.4)^{x}-4=60$$

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range. and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=6^{-x}$$

If the point \((a, b)\) lies on the graph of \(f\) and \(f\) has an inverse, then the point __________ lies on the graph of \(f^{-1}\).

Simplify each expression. (a) \(3^{\log _{3} 7}\) (b) \(4^{\log _{4} 9}\) (c) \(12^{\log _{12} 4}\) (d) \(a^{\log _{\nu} k}(k>0, a>0, a \neq 1)\)

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$5(1.015)^{x-1980}=8$$

Answer each of the following. If \(f(x)=x,\) then for any function \(g\) , $$(f \circ g)(x)=(g \circ f)(x)=$$ __________.

Simplify each expression. (a) \(\log _{3} 3^{19}\) (b) \(\log _{4} 4^{17}\) (c) \(\log _{12} 12^{1 / 3}\) (d) \(\log _{a} \sqrt{a}(a>0, a \neq 1)\)

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$30-3(0.75)^{x-1}=29$$

Answer each of the following. If a function \(f\) has an inverse, then the graph of \(f^{-1}\) may be obtained by reflecting the graph of \(f\) across the line with equation __________.

Graph each function. $$f(x)=\log _{5} x$$

Simplify each expression. (a) \(\log _{3} 1\) (b) \(\log _{4} 1\) (c) \(\log _{12} 1\) (d) \(\log _{a} 1(a>0, a \neq 1)\)

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$5 \ln x=10$$

Answer each of the following. If a function \(f\) has an inverse and \(f(-3)=6,\) then $$f^{-1}(6)=$$ __________.

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$3 \log x=2$$

Sketch the graph of \(f(x)=2^{x}\). Then refer to it and use earlier techniques to graph each function. $$f(x)=2^{x}-4$$

Graph each function. $$f(x)=\log _{1 / 2}(1-x)$$

Evaluate each expression. Do not use a calculator. $$\log 10^{1.5}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (4 x)=1.5$$

Sketch the graph of \(f(x)=2^{x}\). Then refer to it and use earlier techniques to graph each function. $$f(x)=2^{x+1}$$

Answer each of the following. If \(f\) is a function that has an inverse and the graph of \(f\) lies completely within the second quadrant, then the graph of \(f^{-1}\) lies completely within the __________ quadrant.

Graph each function. $$f(x)=\log _{10}(3-x)$$

Evaluate each expression. Do not use a calculator. $$\log 10^{4.3}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (2 x)=5$$

Sketch the graph of \(f(x)=2^{x}\). Then refer to it and use earlier techniques to graph each function. $$f(x)=2^{x-4}$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=3 x-7, \quad g(x)=\frac{x+7}{3}$$

Graph each function. $$f(x)=\log _{3}(x-1)$$

Evaluate each expression. Do not use a calculator. $$\log 10^{\sqrt{3}}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log (2-x)=0.5$$

Sketch the graph of \(f(x)=\left(\frac{1}{3}\right)^{x}\). Then refer to it and use earlier techniques to graph each finction. $$f(x)=\left(\frac{1}{3}\right)^{x}-2$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=4 x+3, \quad g(x)=\frac{x-3}{4}$$

Graph each function. $$f(x)=\log _{2}\left(x^{2}\right)$$

Evaluate each expression. Do not use a calculator. $$\log 10^{\sqrt{3}}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (1-x)=\frac{1}{2}$$