91Ó°ÊÓ

Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(g \circ f)(x)=x^{2 / 3}+3$$

Without using a calculator, evaluate, if possible, the following expressions. $$\sin ^{-1}(-1)$$

Solve the following equations. $$2^{x}=55$$

Without using a calculator, evaluate, if possible, the following expressions. $$\cos \left(\cos ^{-1}(-1)\right)$$

Determine whether the following statements are true and give an explanation or a counterexample. a. All polynomials are rational functions, but not all rational functions are polynomials. b. If \(f\) is a linear polynomial, then \(f \circ f\) is a quadratic polynomial. c. If \(f\) and \(g\) are polynomials, then the degrees of \(f \circ g\) and \(g \circ f\) are equal. d. To graph \(g(x)=f(x+2),\) shift the graph of \(f\) two units to the right.

Solve the following equations. $$5^{3 x}=29$$

Use the table to evaluate the given compositions. $$\begin{array}{lrrrrrr} \hline \boldsymbol{x} & -1 & 0 & 1 & 2 & 3 & 4 \\ \boldsymbol{f}(\boldsymbol{x}) & 3 & 1 & 0 & -1 & -3 & -1 \\ g(\boldsymbol{x}) & -1 & 0 & 2 & 3 & 4 & 5 \\ \boldsymbol{h}(\boldsymbol{x}) & 0 & -1 & 0 & 3 & 0 & 4 \\ \hline \end{array}$$ a. \(h(g(0))\) b. \(g(f(4))\) c. \(h(h(0))\) d. \(g(h(f(4)))\) e. \(f(f(f(1)))\) f. \(h(h(h(0)))\) j. \(f(f(h(3)))\)

Without using a calculator, evaluate, if possible, the following expressions. $$\cos ^{-1}(\cos 7 \pi / 6)$$

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabola \(y=x^{2}+2\) and the line \(y=x+4\)

Simplify the difference quotients\(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) for the following functions. $$f(x)=x^{2}$$