Chapter 2.6, Problem 2E

By definition, (f ○ g)(x) = _____. So if g(2) = 5 and f(5) = 12, then (f ○ g)(2) = _____.

To determine

To evaluate: The value of $\left(f\circ g\right)\left(x\right)$ and the value of $\left(f\circ g\right)\left(2\right)$ if $g\left(2\right)$ is 5 and $f\left(5\right)$ is 12.

Answer to Problem 2E

The value of $\left(f\circ g\right)\left(x\right)$ is $f\left(g\left(x\right)\right)$ and the value of $\left(f\circ g\right)\left(2\right)$ is 12.

Explanation of Solution

The function $f\circ g$ is called the compositions of functions f and g and the domain of $f\circ g$ is set of all x in the domain of g such that $g\left(x\right)$ is in the domain of f, that means,

$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Thus, by above definition, the value of $\left(f\circ g\right)\left(x\right)$ is $f\left(g\left(x\right)\right)$.

Calculation:

From Section1,

$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ (1)

Substitute 2 for x in above formula,

$\left(f\circ g\right)\left(2\right)=f\left(g\left(2\right)\right)$

The value of $g\left(2\right)$ is 5, substitute 5 for $g\left(2\right)$ in above values,

$\left(f\circ g\right)\left(2\right)=f\left(5\right)$

The value of $f\left(5\right)$ is 12, substitute 12 for $f\left(5\right)$ in above values,

$\left(f\circ g\right)\left(2\right)=12$

Thus, the value of $\left(f\circ g\right)\left(2\right)$ is 12.