How does $f(x)=\log _{x}(2)$ compare with $g(x)=\log _{2}(x) ?$ Here is one way to find out. a. Use the equation $\log _{a} b=(\ln b) /(\ln a)$ to express f(x) and g(x) in terms of natural logarithms. b. Graph f and g together. Comment on the behavior of f in relation to the signs and values of g.

Each Exercise gives a formula for a function y = f(x). In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}.$ As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$ $f(x)=(1 / 2) x-7 / 2$

Use graphs to find approximate solutions. $3^{x}-0.5=0$

Graph y = sin x and y = csc x together for $-\pi \leq x \leq 2 \pi.$ Comment on the behavior of csc x in relation to the signs and values of sin x.

Find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. $y=\cos \left(\frac{x}{50}\right)$

Graph y = cos x and y = sec x together for $-3 \pi / 2 \leq x \leq 3 \pi / 2.$ Comment on the behavior of sec x in relation to the signs and values of cos x.

Graph the function.

$$F(x)=\left\{\begin{array}{ll} 4-x^{2}, & x \leq 1 \\ x^{2}+2 x, & x>1 \end{array}\right.$$

Tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $y=x^{3}$ Left 1, down 1

Say whether the function is even, odd, or neither. Give reasons for your answer.

$g(x)=x^3+x$

Each Exercise gives a formula for a function y = f(x). In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}.$ As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$ $f(x)=x^{3}+1$

Find the range. $y=x^{2 / 5}$

The quotient $\left(\log _{4} x\right) /\left(\log _{2} x\right)$ has a constant value. What value? Give reasons for your answer.

Find the domain. $y=x^{2 / 5}$

Use graphs to find approximate solutions. $e^{x}=4$