Given:
The function is:
[tex]f(x)=\dfrac{2x}{3}-4[/tex]
To find:
The inverse of the given function, i.e., [tex]f^{-1}(x)[/tex].
Solution:
We have,
[tex]f(x)=\dfrac{2x}{3}-4[/tex]
Step 1: Substitute [tex]f(x)=y[/tex].
[tex]y=\dfrac{2x}{3}-4[/tex]
Step 2: Interchange x and y.
[tex]x=\dfrac{2y}{3}-4[/tex]
Step 3: Isolate y.
[tex]x+4=\dfrac{2y}{3}[/tex]
[tex]3(x+4)=2y[/tex]
[tex]\dfrac{3x+12}{2}=y[/tex]
[tex]\dfrac{3}{2}x+6=y[/tex]
Step 4: Substitute [tex]y=f^{-1}(x)[/tex].
[tex]f^{-1}(x)=\dfrac{3}{2}x+6[/tex]
Therefore, the inverse of the given function is [tex]f^{-1}(x)=\dfrac{3}{2}x+6[/tex].
