Previously, we have written equations for the inverse of a function. Let’s revisit what we know about inverse functions:
Let’s look at the inverse of an exponential function of the form \(f\left(x\right)=b^x\). Move either the slider \(p\) or the point on the graph. What do you notice? What do you wonder?
The point on the graph has been reflected in the line \(y=x\). Note that the segment perpendicular to the line \(y=x\) (connecting the points on the function \(f\left(x\right)=b^x\) and its inverse) is bisected by the line \(y=x\). To see the inverse function, click on the circle next to the equation \(g\left(y\right)=f\left(y\right)\).
- Inverse functions are reflections of the graph over the line \( y = x \).
- The \((x, y)\) coordinates of inverse functions are switched.
- The domain of the original function becomes the range of the inverse function.
- The range of the original function becomes the domain of the inverse function.
- If two functions \(f\left(x\right)\) and \(g\left(x\right)\) are inverses, then \(f\left(g\left(x\right)\right)=x\) and \(g\left(f\left(x\right)\right)=x\) (which is the composite rule, also known as the identity)
- If the inverse of a function is also a function, the function is said to be one-to-one (also known as injective). Remember, for a function to be one-to-one, the graph of the original function will pass both the vertical and horizontal line tests.
Let’s look at the inverse of an exponential function of the form \(f\left(x\right)=b^x\). Move either the slider \(p\) or the point on the graph. What do you notice? What do you wonder?
The point on the graph has been reflected in the line \(y=x\). Note that the segment perpendicular to the line \(y=x\) (connecting the points on the function \(f\left(x\right)=b^x\) and its inverse) is bisected by the line \(y=x\). To see the inverse function, click on the circle next to the equation \(g\left(y\right)=f\left(y\right)\).
Quick Check
1) What is the domain of the inverse function? 2) What is the range of the inverse function? 3) Is there an asymptote? (Remember, an asymptote is a line that the graph approaches but will not intersect) 4) Is there a y-intercept? 5) What is the x-intercept? Quick Check Solutions |
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The inverse of an exponential function \(f\left(x\right)=b^x\) is the logarithmic function \(f\left(x\right)=\log_bx\).
Graph Inverse Functions
Write Inverse Functions