Exam 9: Logarithmic and Exponential Functions

Graph f(x). State the domain, range, and horizontal asymptote of the function - $f(x)=4^{x}-4$

For the given function $g(x)$ , find a function $f(x)$ such that $(f \circ g)(x)=x$ . - $g(x)=x+5$

Rewrite in exponential form. - $\log _{3}(9 x-4)=1$

Determine whether the graph is the graph of a function. -

Determine whether the functions $f(x)$ and $g(x)$ are inverse functions. - $f(x)=x+3, g(x)=x-3$

For the given function $g(x)$ , find a function $f(x)$ such that $(f \circ g)(x)=x$ . - $g(x)=4 x+7$

Rewrite as a single logarithm using the product rule for logarithms. A ssume all variables represent positive real numbers. - $\log _{7} \mathrm{x}+\log _{7} \mathrm{y}$

Solve the equation. - $\log _{7}(6 x-1)+\log _{7} \mathrm{x}=1$

A fixed point of a function $f(x)$ is a value for which $f(x)=x$ . For the given function, find: a) Fixed point of $f(x)$ b) (f $\circ f)(\mathbf{x})$ c) Fixed point of (f $\circ\mathbf{f})(\mathbf{x})$ - $f(x)=-\frac{1}{2} x+6$

Graph. State the domain, range, and vertical asymptote of the function - $f(x)=\ln (x+3)$

Expand. A ssume that all variables represent positive real numbers. - $\log _{7}\left(\frac{12 \sqrt{\mathrm{r}}}{\mathrm{s}}\right)$

The number of reports of a certain virus has increased exponentially since 1960. The number of cases can be approximated using the function $r(t)=207 \mathrm{e}^{0.006 t}$ , where $t$ is the number of years since 1960. Estimate the number of cases in the year 2000.

Solve. - $4^{x}=\frac{1}{64}$

For the given exponential function, find a) the average rate of change for the interval $[3,5]$ and b) the interval $[3.5,4.5]$ , as well as c) the rate of change at $x=4$ . Round all calculations to the nearest hundredth. - $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}-3}-5$

For the given function $f(x)$ , find $f^{-1}(x)$ . State the domain of $f^{-1}(x)$ . - $f(x)=\sqrt{4 x+5}$

Evaluate. Round to the nearest thousandth, if necessary. - $\log 285$

Evaluate using the change-of-base formula. Round to four decimal places. - $\log _{8.5} 137$

Graph f(x). State the domain, range, and horizontal asymptote of the function - $f(x)=3^{x+3}$

Find the specified domain -For $g(x)=\sqrt{x+3}$ and $h(x)=\frac{1}{x-4}$ , what is the domain of $h \circ g$ ?

Solve. - $3^{1+2 \mathrm{x}}=243$