Exercise 42 Page 269

The graph of a function f(x) and the graph of its inverse function f^(-1)(x) are symmetric with respect to the line y=x. First we will graph the function we choose as f(x). After that we will find and graph its inverse. Finally we will verify the relationship between them using the graph of y=x.

Graphing f(x)

Let's arbitrarily choose f(x)=2x+1 as our function. We can tell that the y-intercept is 1 and the slope is 2. Let's graph this linear function.

Finding and Graphing f^(- 1)(x)

f(x)=2x+1 ⇔ y=2x+1 Next we will switch x and y.

Original equation	Interchange y and x
y=2x+1	x=2y+1

Now let's solve the new equation for y to find the equation of the inverse.

x=2y+1

▼

Solve for y

SubEqn

LHS-1=RHS-1

x-1=2y

DivEqn

.LHS /2.=.RHS /2.

x-1/2=y

RearrangeEqn

Rearrange equation

y=x-1/2

WriteDiffFrac

Write as a difference of fractions

y=x/2-1/2

MoveNumRight

a/b=1/b* a

y=1/2x-1/2

Finally, we replace y with f^(-1)(x) to get the equation of the inverse function. f^(-1)(x)=1/2x-1/2 Let's graph it on the same coordinate plane! We can tell that the slope is 12 and the y-intercept is - 12.

Verifying the Relationship

Finally, let's graph the line y=x.

We can see that the graphs are symmetric with respect to the line y=x. Therefore, they are inverses of each other.