Section 1
Inverse Functions
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=x-1$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=2 x-1$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=\sqrt{x-1}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=-\sqrt{x-1}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=x^{3}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=1+\sqrt[3]{x}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=x^{2}, \quad x \leq 0$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=(1-2 x)^{3}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=\frac{1}{x+1}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=\frac{x}{1+x}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=\frac{1-2 x}{1+x}$$
Show that the functions $f$ are one-to-one, and calculate the inverse functions $f^{-1} .$ Specify the domains and ranges of $f$ and $f^{-1}.$$$f(x)=\frac{x}{\sqrt{x^{2}+1}}$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$g(x)=f(x)-2$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$h(x)=f(2 x)$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$k(x)=-3 f(x)$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$m(x)=f(x-2)$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$p(x)=\frac{1}{1+f(x)}$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$q(x)=\frac{f(x)-3}{2}$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$r(x)=1-2 f(3-4 x)$$
F is a one-to-one function with inverse $f^{-1}$ Calculate the inverses of the given functions in terms of $f^{-1}.$$$s(x)=\frac{1+f(x)}{1-f(x)}$$
Show that the given function is one-to-one and find its inverse.$$f(x)=\left\{\begin{array}{ll}x^{2}+1 & \text { if } x \geq 0 \\x+1 & \text { if } x<0\end{array}\right.$$
Show that the given function is one-to-one and find its inverse.$$h(x)=x|x|+1$$
$$\text { Find } f^{-1}(2) \text { if } f(x)=x^{3}+x$$
Find $g^{-1}(1)$ if $g(x)=x^{3}+x-9$
Find $h^{-1}(-3)$ if $h(x)=x|x|+1$
Assume that the function $f(x)$ satisfies $f^{\prime}(x)=\frac{1}{x}$ and that $f$ is one-to-one. If $y=f^{-1}(x),$ show that $d y / d x=y.$
Find $\left(f^{-1}\right)^{\prime}(x)$ if $f(x)=1+2 x^{3}.$
Show that $f(x)=\frac{4 x^{3}}{x^{2}+1}$ has an inverse and find $\left(f^{-1}\right)^{\prime}(2).$
Find $\left(f^{-1}\right)^{\prime}(-2)$ if $f(x)=x \sqrt{3+x^{2}}.$
If $f(x)=x^{2} /(1+\sqrt{x}),$ find $f^{-1}(2)$ correct to 5 decimal places.
If $g(x)=2 x+\sin x,$ show that $g$ is invertible, and find $g^{-1}(2)$ and $\left(g^{-1}\right)^{\prime}(2)$ correct to 5 decimal places.
Show that $f(x)=x \sec x$ is one-to-one on $(-\pi / 2, \pi / 2)$ What is the domain of $f^{-1}(x) ?$ Find $\left(f^{-1}\right)^{\prime}(0).$
If functions $f$ and $g$ have respective inverses $f^{-1}$ and $g^{-1}$ show that the composite function $f \circ g$ has inverse $(f \circ g)^{-1}=g^{-1} \circ f^{-1}$
Can an even function be self-inverse? an odd function?
In this section it was claimed that an increasing (or decreasing) function defined on a single interval is necessarily one-to-one. Is the converse of this statement true? Explain.
Repeat Exercise 37 with the added assumption that $f$ is continuous on the interval where it is defined.