Chapter 5.1, Problem 6AYU

True or False The domain of the composite function $(f°g)(x)$ is the same as the domain of $g(x)$.

To determine

To find: The domain of the composite function $(f\circ g)\left(x\right)$ is the same as the domain of $g\left(x\right)$.

Check whether given statement is True or False.

Answer to Problem 6AYU

Solution:

False

$\text{The domain of the composite function }\left(f\circ g\right)\left(x\right)\text{is not the same as the domain of }g\left(x\right)$.

$\text{The domain of the composite function }\left(f\circ g\right)\left(x\right)\text{will be the range of the function }g\left(x\right)$.

Explanation of Solution

Given:

The domain of the composite function $(f\circ g)\left(x\right)$ is the same as the domain of $g\left(x\right)$.

Calculation:

In general, suppose that $f$ and $g$ are two functions and that $x$ is a number in the domain of $g$, evaluating $g$ at $x$ yields $g\left(x\right)$.

If $g\left(x\right)$ is in the domain of $f$, then evaluating $f$ at $g\left(x\right)$ yields the expression $f\left(g\left(x\right)\right)$.

The correspondence from $x$ to $f\left(g\left(x\right)\right)$ is called a a composite function i.e., $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$.

We also see that $g\left(x\right)$ must be a member of the domain of $f$ otherwise the second function evaluation in $f\left(g\left(x\right)\right)$ cannot be completed, and the expression is still undefined.

Thus the domain of $\left(f\circ g\right)$ consists of only those inputs in the domain of $g$ that produce outputs from $g$ belonging to the domain of $f$.

Note that the domain of $f$ composed with $g$ is the set of all $x$ such that $x$ is in the domain of $g$ and $g\left(x\right)$ is in the domain of $f$.