Question

find the inverse of each of the given functions. ( f(x)=4x - 12 ), ( f^{-1}(x)=square x+square ). ( h(x)=\frac{2x - 4}{3} ). ( h^{-1}(x)=\frac{3x + 4}{2} ) ( h^{-1}(x)=\frac{(2x - 4)}{3} ) ( h^{-1}(x)=\frac{3x - 12}{2} )

Explanation:

Step1: Let \(y = f(x)\)

Let \(y = 4x-12\).

Step2: Swap \(x\) and \(y\)

We get \(x = 4y - 12\).

Step3: Solve for \(y\)

Add 12 to both sides: \(x + 12=4y\). Then divide both sides by 4, so \(y=\frac{x + 12}{4}=\frac{1}{4}x+3\). So \(f^{-1}(x)=\frac{1}{4}x + 3\).

For \(h(x)=\frac{2x - 4}{3}\):

Step1: Let \(y = h(x)\)

Let \(y=\frac{2x - 4}{3}\).

Step2: Swap \(x\) and \(y\)

We have \(x=\frac{2y - 4}{3}\).

Step3: Solve for \(y\)

Multiply both sides by 3: \(3x=2y - 4\). Add 4 to both sides: \(3x + 4=2y\). Then \(y=\frac{3x + 4}{2}\). So \(h^{-1}(x)=\frac{3x + 4}{2}\).

Answer:

For \(f(x)=4x - 12\), \(f^{-1}(x)=\frac{1}{4}x+3\); for \(h(x)=\frac{2x - 4}{3}\), \(h^{-1}(x)=\frac{3x + 4}{2}\)