91Ó°ÊÓ

Prove that if \(f\) and \(g\) are one-to-one functions, then \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x) .\)

Find the difference quotient and simplify your answer. $$ f(x)=x^{3}+3 x, \quad \frac{f(x+h)-f(x)}{h}, \quad h \neq 0 $$

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correct- that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. (a) \(y=x-8\) (b) \(y=x+1\) (c) \(y=-x+3\)

Find the difference quotient and simplify your answer. $$ f(x)=4 x^{2}-2 x, \quad \frac{f(x+h)-f(x)}{h}, \quad h \neq 0 $$

Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

Find a relationship between \(x\) and \(y\) such that \((x, y)\) is equidistant (the same distance) from the two points. $$ (4,-1),(-2,3) $$

Find the difference quotient and simplify your answer. $$ g(x)=\frac{1}{x^{2}}, \frac{g(x)-g(3)}{x-3}, \quad x \neq 3 $$

Find the difference quotient and simplify your answer. $$ f(t)=\frac{1}{t-2}, \quad \frac{f(t)-f(1)}{t-1}, \quad t \neq 1 $$

Find the difference quotient and simplify your answer. $$ f(x)=\sqrt{5 x}, \frac{f(x)-f(5)}{x-5}, x \neq 5 $$

Determine if the situation could be represented by a one-to-one function. If so, write a statement that describes the inverse function. The number of miles \(n\) a marathon runner has completed in terms of the time \(t\) in hours