Exercise 18 Page 326

Before we can find the inverse of the given function, we need to replace f(x) with y. f(x)=(1/9)^x ⇔ y=(1/9)^x To find the inverse of a function, we start by exchanging the variables in the given function. Then we can solve for y. Note that the inverse of an exponential function is a logarithmic function. ccc Given Function & & Inverse Function y=(1/9)^x & change ⟶ & x=(1/9)^yNow we have to isolate y in the inverse equation. To do this, we can use the definition of a logarithm to rewrite the equation. ( 1/9)^y= x ⇔ y=log_(19) x Now we have the inverse of the given function. y=log_()19 x

Graphing the Function

We want to draw a graph of the given function. y=(1/9)^x Because the base of the function is greater than 0, but less than 1, we know that this is an exponential decay function. To draw the graph, we will start by making a table of values.

The ordered pairs ( - 1, 9), ( 0, 1), ( 1, and 19) all lie on the graph of the function. Now, we will plot and connect these points with a smooth curve.

Graphing the Inverse of the Function

Finally, we can graph the inverse of the function by reflecting the parabola across the line y=x. This means that we should interchange the x- and y-coordinates of the points that are on the graph of the given function.