Inverse functions undo each other. If a point \((a,b)\) lies on the graph of \(f\), then the point
\((b,a)\) lies on the graph of \(f^{-1}\).
\[
(a,b)\mapsto(b,a)
\]
This coordinate swap is why the graphs of a function and its inverse are mirror images over the line:
\[
y=x
\]
1. How to find an inverse
The basic steps are the same for exponential and logarithmic functions:
- Write \(y=f(x)\).
- Swap \(x\) and \(y\).
- Solve the new equation for \(y\).
- Replace \(y\) with \(f^{-1}(x)\).
- Swap the domain and range.
2. Exponential function inverse
Start with a transformed exponential function:
\[
f(x)=A\cdot b^{k(x-h)}+v
\]
Let \(y=f(x)\):
\[
y=A\cdot b^{k(x-h)}+v
\]
Swap \(x\) and \(y\):
\[
x=A\cdot b^{k(y-h)}+v
\]
Isolate the exponential part:
\[
\frac{x-v}{A}=b^{k(y-h)}
\]
Convert exponential form to logarithmic form:
\[
k(y-h)=\log_b\left(\frac{x-v}{A}\right)
\]
Solve for \(y\):
\[
y=h+\frac{1}{k}\log_b\left(\frac{x-v}{A}\right)
\]
Therefore:
\[
f^{-1}(x)=h+\frac{1}{k}\log_b\left(\frac{x-v}{A}\right)
\]
3. Logarithmic function inverse
Start with a transformed logarithmic function:
\[
f(x)=A\log_b\left(k(x-h)\right)+v
\]
Let \(y=f(x)\):
\[
y=A\log_b\left(k(x-h)\right)+v
\]
Swap \(x\) and \(y\):
\[
x=A\log_b\left(k(y-h)\right)+v
\]
Isolate the logarithm:
\[
\log_b\left(k(y-h)\right)=\frac{x-v}{A}
\]
Convert logarithmic form to exponential form:
\[
k(y-h)=b^{(x-v)/A}
\]
Solve for \(y\):
\[
y=h+\frac{1}{k}b^{(x-v)/A}
\]
Therefore:
\[
f^{-1}(x)=h+\frac{1}{k}b^{(x-v)/A}
\]
4. Domain and range switch
A function and its inverse switch domain and range.
\[
\text{Domain of }f^{-1}=\text{Range of }f
\]
\[
\text{Range of }f^{-1}=\text{Domain of }f
\]
For an exponential function, the original domain is all real numbers. Its range is controlled by the
horizontal asymptote \(y=v\).
- If \(A>0\), the exponential range is \(y>v\).
- If \(A<0\), the exponential range is \(y
That range becomes the domain of the logarithmic inverse.
5. Asymptote reflection
Asymptotes also reflect over the line \(y=x\).
A transformed exponential function has horizontal asymptote:
\[
y=v
\]
Its inverse has vertical asymptote:
\[
x=v
\]
A transformed logarithmic function has vertical asymptote:
\[
x=h
\]
Its inverse has horizontal asymptote:
\[
y=h
\]
6. Worked exponential example
Find the inverse of:
\[
f(x)=4\cdot3^{x-2}
\]
Write \(y=f(x)\):
\[
y=4\cdot3^{x-2}
\]
Swap \(x\) and \(y\):
\[
x=4\cdot3^{y-2}
\]
Isolate the power:
\[
\frac{x}{4}=3^{y-2}
\]
Convert to logarithmic form:
\[
\log_3\left(\frac{x}{4}\right)=y-2
\]
Solve for \(y\):
\[
y=\log_3\left(\frac{x}{4}\right)+2
\]
Therefore:
\[
\boxed{f^{-1}(x)=\log_3\left(\frac{x}{4}\right)+2}
\]
7. Worked logarithmic example
Find the inverse of:
\[
f(x)=\log_3(x-2)+4
\]
Write \(y=f(x)\):
\[
y=\log_3(x-2)+4
\]
Swap \(x\) and \(y\):
\[
x=\log_3(y-2)+4
\]
Isolate the logarithm:
\[
x-4=\log_3(y-2)
\]
Convert to exponential form:
\[
3^{x-4}=y-2
\]
Solve for \(y\):
\[
y=3^{x-4}+2
\]
Therefore:
\[
\boxed{f^{-1}(x)=3^{x-4}+2}
\]
8. One-to-one requirement
A function must be one-to-one to have an inverse function. Exponential and logarithmic functions with
valid parameters are one-to-one because they are monotonic.
- The base must satisfy \(b>0\) and \(b\neq1\).
- The vertical scale must satisfy \(A\neq0\).
- The horizontal multiplier must satisfy \(k\neq0\).
If \(A=0\) or \(k=0\), the function loses the needed one-to-one behavior.
9. Composition check
A correct inverse satisfies two composition checks:
\[
f(f^{-1}(x))=x
\]
\[
f^{-1}(f(x))=x
\]
These equations mean that the function and inverse undo each other.
10. Summary table
| Original function |
Inverse function |
Main conversion |
| \(f(x)=A\cdot b^{k(x-h)}+v\) |
\(f^{-1}(x)=h+\frac{1}{k}\log_b\left(\frac{x-v}{A}\right)\) |
Exponential form changes to logarithmic form. |
| \(f(x)=A\log_b(k(x-h))+v\) |
\(f^{-1}(x)=h+\frac{1}{k}b^{(x-v)/A}\) |
Logarithmic form changes to exponential form. |
| Point \((a,b)\) on \(f\) |
Point \((b,a)\) on \(f^{-1}\) |
Coordinates swap. |
| Horizontal asymptote \(y=v\) |
Vertical asymptote \(x=v\) |
Reflection over \(y=x\). |
| Vertical asymptote \(x=h\) |
Horizontal asymptote \(y=h\) |
Reflection over \(y=x\). |
