Composition and Inverses

Example 1

For $\displaystyle f(x)=2x+1$, $\displaystyle g(x)=x^2$, find $\displaystyle f(g(3))$.

$\displaystyle g(3)=9$, then $\displaystyle f(9)=19$. Reversing the order gives a different expression.

Example 2

Find the inverse of $\displaystyle f(x)=3x-4$.

Write $\displaystyle y=3x-4$, swap roles, and solve $\displaystyle x=3y-4$. Thus $\displaystyle f^{-1}(x)=(x+4)/3$.

Problem 1.3.1

For $\displaystyle f(x)=x+2$ and $\displaystyle g(x)=3x$, find $\displaystyle (g\circ f)(x)$.

Problem 1.3.2

Find the inverse of $\displaystyle h(x)=(x-5)/2$.

Problem 1.3.3

The graph shows $\displaystyle y=x^2$ restricted to $\displaystyle x\ge0$ and its inverse. State the inverse and explain the restriction.

Show worked solution

The inverse is $\displaystyle y=\sqrt{x}$, with domain and range $\displaystyle [0,\infty)$. Restricting the parabola to $\displaystyle x\ge0$ makes it one-to-one; the graphs reflect across $\displaystyle y=x$.