Exam 10: Compositions, Inverses, and Combinations of Functions

Given $f(x)=\frac{5+\ln (x-3)}{5-\ln (x-3)}$ , does $f^{-1}(x)=e^{\frac{5 x-5}{x+1}}+3$ ?

Based on the following table, could the function $f(x)$ be invertible?

Select the list of functions that is a decomposition of $f(x)=\csc ^{3}(\ln x)$ in the form of $g(h(p(x)))$ .

Is the quotient of two even functions; odd, even, or neither?

The functions $m(x)$ and $n(x)$ are defined by the graph below. The dashed graph is $m(x)$ and the solid graph is $n(x)$ . If $p(x)=-2 m(x+3)+6$ , what is $p(-2)$ ?

Find the function $p(x)=f(x)+g(x)$ when $f(x)=4 x^{3}+5$ and $g(x)=3 x+2$ .

Suppose $f(x)=\sqrt{8-x}$ . What is the domain of $f^{-1}(x)$ ?

Given $f(x)=\frac{5}{x}$ and $g(x)=x^{2}-1$ , what is $g(f(x)$ ?

Let $f(x)=\sqrt{a x+1}$ and $g(x)=\ln (b x)$ for positive constants $a$ and $b$ . Let $h(x)=a x \ln \left(b^{2} x^{2}\right)+\ln \left(b^{2} x^{2}\right)$ . If $f(4)=2$ and $g(4)=3$ , what is $h(4) ?$

Let $f(x)=x+8$ and $h(x)=\sqrt{x-5}$ . What is $(h(f(x)))^{2}$ ?

Let $f(x)=x^{2}+5$ . Does $f(x+h)=x^{2}+h^{2}+5$ ?

Let $P=20 \ln (t)$ give the annual profit of a company (in thousands of dollars) $t$ years after its formation. What is $P^{-1}(38)$ ? Round to the nearest whole number and include units.

Given the graphs of $f$ and $g$ below, evaluate $f(g(-3))$ .

Is the function $f(x)=\frac{x^{3}+4}{4}$ the inverse of the function $g(t)=\sqrt[3]{4 t-4}$ ?

Let $f(x)=\frac{1}{7+x}$ . Does $f^{-1}(x)=\frac{1+7 x}{x}$ ?

Find $f(x)(g(x))^{2}$ if $f(x)=5 e^{2 x}+1$ and $g(x)=\frac{1}{7} e^{-x}$ .

Let $f(x)=\left(x^{2}+1\right)(x+2)$ and $g(x)=(x-1)\left(x^{2}-4\right)$ . Find a simplified formula for $h(x)=\frac{5 f(x)}{17 g(x)}$ .

For positive numbers $x$ , what is the inverse of $h(x)=e^{\sqrt{x}-2}$ ?

Let $f(x)=x^{2}+5$ and $g(x)=\frac{1}{x}+2$ . What is $f(4)+g(1)$ ?

If $u(v(x))=\frac{4}{1+\sqrt{x}}$ , which of the following could be true? Mark all that apply.