Determine whether or not the function determined by the given equation is one-to-one.

$$f(x)=2 x-3$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$G(x)=-3 x+7$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$h(x)=2 x+5$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$w(x)=-7 x+9$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$f(x)=2 x^{2}-7$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$g(x)=-4 x^{2}+1$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$h(x)=\sqrt{2 x+3}$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$r(x)=\sqrt{2-5 x}$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$v(x)=4 x^{2}+1, x \geq 0$$

Bobby B.

University of North Texas

Determine whether or not the function determined by the given equation is one-to-one.

$$s(x)=-2 x^{2}+6, x \leq 0$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$f(x)=2 x-3$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$G(x)=-3 x+7$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$h(x)=2 x+5$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$w(x)=-7 x+9$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$f(x)=2 x^{2}-7$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$g(x)=-4 x^{2}+1$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$h(x)=\sqrt{2 x+3}$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$r(x)=\sqrt{2-5 x}$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$v(x)=4 x^{2}+1, x \geq 0$$

Bobby B.

University of North Texas

Determine if the equation describes an increasing or decreasing function or neither.

$$s(x)=-2 x^{2}+6, x \leq 0$$

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

Determine if the given graph represents a one-to-one function.

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=2 x-3$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=-3 x+7$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=\sqrt{2 x-8}$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=\sqrt{6-2 x}$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=x^{2}, x \geq 0$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=2 x^{2}+1, x \geq 0$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=-4 x^{2}+7, x \leq 0$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=8 x^{3}$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=2 \sqrt[3]{x}$$

Bobby B.

University of North Texas

In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.

$$f(x)=(x-1)^{3}$$

Bobby B.

University of North Texas

If $f(x)=-3 x+9,$ find $\left(\text { a) } f^{-1}(2), \text { (b) } f^{-1}(5)\right.$

Bobby B.

University of North Texas

If $f(x)=\sqrt{2 x-1},$ find $(a) f^{-1}(1)$,

(b) $f^{-1}(3)$

Bobby B.

University of North Texas

If $f(x)=\sqrt{3 x+2},$ find (a) $f^{-1}(3),$ (b) $f^{-1}(4)$

Bobby B.

University of North Texas

If $f(x)=\frac{2 x+1}{3 x+2},$ find $\left(\text { a) } f^{-1}(0), \text { (b) } f^{-1}(-1)\right.$

Bobby B.

University of North Texas

If $f(x)=\frac{2 x-3}{3 x+4},$ find (a) $f^{-1}(-3),$ (b) $f^{-1}(3)$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=5 x-9$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=2 x+1$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=2 x^{2}+3, x \geq 0$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=-5 x^{2}+3, x \leq 0$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=\sqrt{2 x+3}$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=-\sqrt{5-4 x}$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=\frac{2 x+7}{9 x-3}$$

Bobby B.

University of North Texas

Determine the equation of the inverse function.

$$f(x)=\frac{11-3 x}{2 x+5}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=2 x+5, g(x)=\frac{x-5}{2}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=3 x-7, g(x)=\frac{x+7}{3}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=5 x+9, g(x)=\frac{x-9}{5}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=2 x-3, g(x)=\frac{x+3}{2}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=\sqrt{9-x^{2}}, g(x)=\sqrt{9-x^{2}}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=\sqrt{5-x^{2}}, g(x)=\sqrt{5-x^{2}}$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=\sqrt{2 x-3}, g(x)=1 / 2 x^{2}+3 / 2 x \geq 0$$

Bobby B.

University of North Texas

Verify that $f$ and $g$ are inverse functions using the composition property.

$$f(x)=x^{2}+1, x \geq 0, g(x)=\sqrt{x-1}$$

Bobby B.

University of North Texas

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.

Exercise 29

Bobby B.

University of North Texas

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.

Exercise 32

Bobby B.

University of North Texas

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.

Exercise 35

Bobby B.

University of North Texas

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.

Exercise 37

Bobby B.

University of North Texas

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.

$$f(x)=\frac{3 x-2}{2 x+5}$$

Bobby B.

University of North Texas

(a) Given $f(x)=2 x^{3}+3 x-4,$ show this function is one-to-one

(b) Determine $\left(f^{-1}(x)\right)^{\prime}(18)$.

Bobby B.

University of North Texas

(a) Given $f(x)=3 x^{5}+2 x^{3}+2,$ show this function is one-to-one.

(b) Determine $\left(f^{-1}(x)\right)^{\prime}(7)$.

Bobby B.

University of North Texas

Prove, using (4) that $\frac{d}{d x}\left(x^{1 / 3}\right)=\frac{1}{3 x^{2 / 3}}$.

Bobby B.

University of North Texas

Consider the function defined by

$$y=f(x)=\left\{\begin{array}{cc}x & 0 \leq x < 1 \\ -x & x \geq 1\end{array}\right.$$

sketch the graph of this function and determine if it is one-to-one.

Bobby B.

University of North Texas

Consider the function defined by

$$y=f(x)=\left\{\begin{array}{cc}-2 x & 0 \leq x < 2 \\ 4 x & x \geq 2\end{array}\right.$$

sketch the graph of this function and determine if it is one-to-one.

Bobby B.

University of North Texas

Will a one-to-one function always be a decreasing or increasing function?

Bobby B.

University of North Texas

(a) Do even continuous functions have an inverse? (b) Odd functions?

Bobby B.

University of North Texas

Consider $f(x)=\frac{4 x}{x^{2}-9},$ (a) Show that this function is always decreasing.

(b) Does this function have an inverse? Explain.

Bobby B.

University of North Texas

Suppose $f$ and $g$ are inverse functions and have second derivatives. Show that

$$g^{\prime \prime}(x)=-\frac{f^{\prime \prime}(g(x))}{\left(f^{\prime}(g(x))\right)^{3}}$$

Bobby B.

University of North Texas

Using the results of the previous exercise, determine how the concavity of $g$ is related to the concavity of $f$ Hint: there are four cases to consider.

Check back soon!

Prove the inverse function is unique. Hint: assume both $g$ and $h$ are inverse functions of $f$ consider $(g f h)(x)$.

Bobby B.

University of North Texas