Hello there. To solve this question, we have to remember some properties about determining the inverse of a function.
First, a function is called invertible if it is bijective, that is, it is both injective and surjective.
This inverse is unique, in the sense that for a function f(x), there exists only g(x) such that
[tex]g(x)=f^{-1}(x)[/tex]The property that inverse functions satisfy is
[tex]f(f^{-1}(x))=x[/tex]And we'll use it to find it.
Okay. First, suppose that this function has an inverse g(x), that we'll call as
[tex]g(x)=y[/tex]So using the property on inverse functions, we get
[tex]f(y)=4y-7[/tex]So this might be equal to
[tex]4y-7=x[/tex]Solve the equation for y.
Add 7 on both sides of the equation
[tex]4y=x+7[/tex]Divide both sides of the equation by a factor of 4
[tex]y=\dfrac{x}{4}+\dfrac{7}{4}[/tex]Such that we get
[tex]f^{-1}(x)=\dfrac{x}{4}+\dfrac{7}{4}[/tex]This is the inverse of f.
