Given:
[tex]\begin{gathered} f(x)=13x+2 \\ \\ g(x)=3x^2-13 \\ \\ h(x)=\frac{13}{x+13} \end{gathered}[/tex]
Find-:
The inverse of a function.
Explanation-:
(a)
For the inverse of a function, x change as y and y change as x and solve for 'y'
[tex]\begin{gathered} f(x)=13x+2 \\ \\ f(y)=13y+2 \\ \\ x\rightarrow y \\ \\ y\rightarrow x \\ \\ \end{gathered}[/tex]
Then solve,
[tex]\begin{gathered} y=13x+2 \\ \\ y-2=13x \\ \\ x=\frac{y-2}{13} \end{gathered}[/tex]
So, value,
[tex]f^{-1}(y)=\frac{y-2}{13}[/tex]
(b)
[tex]g(x)=3x^2-13[/tex]
So, the value is:
[tex]g(y)=3y^2-13[/tex]
The inverse of a function is:
[tex]\begin{gathered} x=3y^2-13 \\ \\ x\rightarrow y \\ \\ y\rightarrow x \\ \\ y=3x^2-13 \\ \\ 3x^2=y+13 \\ \\ x^2=\frac{y+13}{3} \\ \\ x=\sqrt{\frac{y+13}{3}} \end{gathered}[/tex]
So, the inverse value is:
[tex]g^{-1}(y)=\sqrt{\frac{y+13}{3}}[/tex]
(c)
[tex]h(x)=\frac{13}{x+13}[/tex]
Value of h(y) is:
[tex]h(y)=\frac{13}{y+13}[/tex]
Then solve for inverse function,
[tex]\begin{gathered} x=\frac{13}{y+13} \\ \\ x\rightarrow y \\ \\ y\rightarrow x \\ \\ y=\frac{13}{x+13} \\ \\ y(x+13)=13 \\ \\ x+13=\frac{13}{y} \\ \\ x=\frac{13}{y}-13 \end{gathered}[/tex]
So, inverse value is:
[tex]h^{-1}(y)=\frac{13}{y}-13[/tex]
