Write Inverse Functions
We will be writing the inverse of exponential functions of the form \(f\left(x\right)=a\cdot b^{x-h}+k\) and the inverse of logarithmic functions of the form \(f\left(x\right)=a\log_b\left(x-h\right)+k\).
Steps to write inverse functions:
1. Switch the \(x\) and \(y\) variables.
2. Isolate \(b^{x-h}\) or \(\log_b\left(x-h\right)\).
3. Rewrite the exponential function as a logarithmic function or the logarithmic function as an exponential function.
4. Solve for \(y\) and write using inverse function notation.
Example 1: Write the inverse of \(f\left(x\right)=3\log_2\left(x-7\right)+1\).
\(\begin{align}&y=3\log_2\left(x-7\right)+1\ & \ \ \ \ \ \ &\text{1) Replace}\ \ f\left(x\right)\ \text{with y.}\\
&x=3\log_2\left(y-7\right)+1\ & \ \ \ \ \ \ &\text{2) Switch the x and y variables.}\\
&x-1=3\log_2\left(y-7\right)\ & \ \ \ \ \ \ &\text{3) Isolate the logarithmic expression (use inverse operations on the constant).}\\
&\frac{x-1}{3}=\log_2\left(y-7\right)\ & \ \ \ \ \ \ &\text{4) Divide both sides by the coefficient.}\\
&2^{\frac{x-1}{3}}=y-7\ & \ \ \ \ \ \ &\text{5) Write the logarithmic equation in its equivalent exponential form.}\\
&2^{\frac{x-1}{3}}+7=y\ & \ \ \ \ \ \ &\text{6) Solve for y.}\\
&f^{-1}\left(x\right)=2^{\frac{x-1}{3}}+7\ & \ \ \ \ \ \ &\text{7) Write using inverse function notation.}\end{align}\)
Example 2: Write the inverse of \(f\left(x\right)=3\left(\frac{1}{2}\right)^{x-1}+4\).
We will be writing the inverse of exponential functions of the form \(f\left(x\right)=a\cdot b^{x-h}+k\) and the inverse of logarithmic functions of the form \(f\left(x\right)=a\log_b\left(x-h\right)+k\).
Steps to write inverse functions:
1. Switch the \(x\) and \(y\) variables.
2. Isolate \(b^{x-h}\) or \(\log_b\left(x-h\right)\).
3. Rewrite the exponential function as a logarithmic function or the logarithmic function as an exponential function.
4. Solve for \(y\) and write using inverse function notation.
Example 1: Write the inverse of \(f\left(x\right)=3\log_2\left(x-7\right)+1\).
\(\begin{align}&y=3\log_2\left(x-7\right)+1\ & \ \ \ \ \ \ &\text{1) Replace}\ \ f\left(x\right)\ \text{with y.}\\
&x=3\log_2\left(y-7\right)+1\ & \ \ \ \ \ \ &\text{2) Switch the x and y variables.}\\
&x-1=3\log_2\left(y-7\right)\ & \ \ \ \ \ \ &\text{3) Isolate the logarithmic expression (use inverse operations on the constant).}\\
&\frac{x-1}{3}=\log_2\left(y-7\right)\ & \ \ \ \ \ \ &\text{4) Divide both sides by the coefficient.}\\
&2^{\frac{x-1}{3}}=y-7\ & \ \ \ \ \ \ &\text{5) Write the logarithmic equation in its equivalent exponential form.}\\
&2^{\frac{x-1}{3}}+7=y\ & \ \ \ \ \ \ &\text{6) Solve for y.}\\
&f^{-1}\left(x\right)=2^{\frac{x-1}{3}}+7\ & \ \ \ \ \ \ &\text{7) Write using inverse function notation.}\end{align}\)
Example 2: Write the inverse of \(f\left(x\right)=3\left(\frac{1}{2}\right)^{x-1}+4\).
Example 3: Use the composite rule to verify that \(f\left(x\right)\) and \(g\left(x\right)\) are inverses of each other.
\(f\left(x\right)=3\log_2\left(x-1\right)\) and \(g\left(x\right)=2^{\frac{x}{3}}+1\)
Don't forget about base \(e\) and natural logarithms when writing inverses! The process is the same but we use \(ln\) instead of \(log\).
Example 4: Find the inverse of \(f\left(x\right)=5e^{2x}\).
\(\begin{align}&y=5e^{2x}\ & \ \ \ \ \ \ &\text{1) Replace}\ \ f\left(x\right)\ \text{with y.}\\
&x=5e^{2y}\ & \ \ \ \ \ \ &\text{2) Switch the x and y variables.}\\
&\frac{x}{5}=e^{2y}\ & \ \ \ \ \ \ &\text{3) Isolate the base (divide by the coefficient).}\\
&\ln\frac{x}{5}=2y\ & \ \ \ \ \ \ &\text{4) Write the exponential equation in its equivalent logarithmic form (remember for base e use ln).}\\
&\frac{1}{2}\ln\frac{x}{5}=y\ & \ \ \ \ \ \ &\text{5) Solve for y.}\\
&f^{-1}\left(x\right)=\frac{1}{2}\ln\left(\frac{x}{5}\right)\ & \ \ \ \ \ \ &\text{6) Write using inverse function notation.}\end{align}\)
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