To find the inverse function, we can follow the next steps:
First Function
1. Replace x with y as follows:
[tex]y=3x+7\Rightarrow x=3y+7[/tex]
2. Solve the resulting equation for y. Subtract 7 from both sides of the equation:
[tex]x-7=3y+7-7\Rightarrow x-7=3y[/tex]
3. Divide both sides of the equation by 3:
[tex]\frac{(x-7)}{3}=\frac{3}{3}y\Rightarrow y=\frac{(x-7)}{3}=\frac{1}{3}(x-7)=\frac{x}{3}-\frac{7}{3}[/tex]Second Function
We need to repeat the process to obtain the inverse of this function:
1. Replace x with y:
[tex]y=8x\Rightarrow x=8y[/tex]
2. Solve for y. Divide both sides by 8:
[tex]\frac{x}{8}=\frac{8}{8}y\Rightarrow y=\frac{x}{8}[/tex]
In summary, we have that the inverse functions are:
For function
[tex]y=3x+7[/tex]
The inverse function is:
[tex]y=F^{-1}^{}(x)=\frac{(x-7)}{3}[/tex]
And, for the function
[tex]y=8x[/tex]
The inverse function is:
[tex]y=f^{-1}(x)=\frac{1}{8}x[/tex]
