Inverse Functions

$f^{-1}(x)$ is defined as the inverse function of $f(x)$ if it consistently reverses the $f(x)$ process. That is, if $f(x)$ turns $a$ into $b$, then $f^{-1}x$ must turn $b$ into $a$. More concisely and formally, $f^{-1}x$ is the inverse function of $f(x)$ if $f({f}^{-1}(x))=x$.

Inverse functions' domain and range

If $f$ maps $X$ to $Y$, then $f^{-1}$ maps $Y$ back to $X$.

Domain Restrictions: Parabola

Informally, a restriction of a function is the result of trimming its domain.   Remember that: If $f$ maps $X$ to $Y$, then $f^{-1}$ maps $Y$ back to $X$.  This is not true of the function $f(x)=x^2$. 

Without any domain restriction, $f(x)=x^2$ does not have an inverse function as it fails the horizontal line test. But if we restrict the domain to be $x > 0$ then we find that it passes the horizontal line test and therefore has an inverse function.  Below is the graph of the parabola and its "inverse."  Notice that the parabola does not have a "true" inverse because the original function fails the horizontal line test and must have a restricted domain to have an inverse.

Failure of horizontal line test

Graph of a parabola with the equation $y=x^2$, the U-Shaped curve opening up. This function fails the horizontal line test, and therefore does not have an inverse. The inverse equation, $y=\sqrt{x}$ (other graph) only includes the positive input values of the parabola's domain. However, if we restrict the domain to be $x>0$, then we find that it passes the horizontal line test and will match the inverse function.

Domain Restriction:  Exponential and Logarithmic Functions 

Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.  The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

Example 1

Is  $x=0$ in the domain of the function $f(x)=log(x)$?  If so, what is the value of the function when $x=0$?  Verify the result.

No, the function has no defined value for $x=0$. To verify, suppose $x=0$ is in the domain of the function $f(x)=log(x)$. Then there is some number $n$ such that $n=log(0)$. Rewriting as an exponential equation gives: $10n=0$, which is impossible since no such real number $n$ exists. Therefore, $x=0$ is not in the domain of the function $f(x)=log(x)$.