Inverse Functions

A function $f$ is one-to-one (1-1) if it does not assign the same value to two different elements of its domain:

If $f(a)=f(b)$, then $a=b$.

If f is a 1-1 function, then it has an inverse function $f^{-1}$ defined by $f^{-1}(y)=x$ iff $f(x)=y$, for all $y$ in the range of f.

The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

To find a formula for $f^{-1}$, we can

1. Set $y=f(x)$.

2. Solve for $x$ in terms of $y$, if possible.

3. Set $f^{-1}(y)=x$.

[Another common way to do this is to

1. Set $y=f(x)$, and then interchange $x$ and $y$.

2. Solve for $y$ in terms of $x$, if possible.

3. Set $f^{-1}(x)=y$.]

Ex 1 Show whether or not the function $f(x)=5x^3+9$ is one-to-one.

Sol $f(a)=f(b) \Rightarrow 5a^3+9=5b^3+9 \Rightarrow 5a^3=5b^3$ $\Rightarrow a^3=b^3 \Rightarrow a=b$, so $f$ is a 1-1 function.

Ex 2 Show whether or not the function $f(x)=x^2-6x+11$ is one-to-one.

Sol Setting $f(x)=11$, for example, and solving gives that $f(0)=11=f(6)$; so $f$ is not a 1-1 function.

Ex 3 If $f(x)=\frac{x}{3}-5$, find a formula for $f^{-1}(x)$

Sol Let $y=\frac{x}{3}-5$. Then $3y=x-15$, so $3y+15=x$ or $x=3y+15$. Thus $f^{-1}(y)=3y+15$, so $f^{-1}(x)=3x+15$.

Pr 1 If $f(x)=5x-8$, find a formula for $f^{-1}(y)$.

Pr 2 If $f(x)=2x^3-5$, find a formula for $f^{-1}(y)$.

Pr 3 If $f(x)=\sqrt{4x-9}$, find a formula for $f^{-1}(y)$ and find the domain for $f^{-1}$.

Pr 4 If $f(x)=\frac{2x-5}{x+4}$, find a formula for $f^{-1}(y)$.

Pr 5 Show whether or not the function $f(x)=x+\frac{4}{x}$ has an inverse.

Pr 6 Let $f(x)=x^2-4x+9$ for $x\ge 2$. Find a formula for $f^{-1}(y)$ and find the domain for $f^{-1}$.

Pr 7 If $f(x)=\frac{x}{x^2-4}$ for $-2<x<2$, find $f^{-1}(1/2)$.

Pr 8 If $f(x)=\frac{8}{\sqrt{x}+2}$, find a formula for $f^{-1}(y)$ and find the domain for $f^{-1}$.

Go to Solutions.

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