Question

Let $f$ be a function with domain $[0,7]$ and $g$ be a function defined by $g(x)=|2x+1|$. Then the domain of $(f\circ g)(x)$ is

• A. $[0,7]$
• B. $[-7,0]$
• C. $[-4,3]$
• D. $[-3,4]$

Hint:

For composition, force the inner function output to lie inside the domain of the outer function.

Answer 1

The Correct Option is C

Step 1: Idea of a composite function.
In $(f\circ g)(x)=f(g(x))$ we first work out $g(x)$, then feed that answer into $f$. So the output of $g$ must be a value that $f$ is allowed to receive.
Step 2: What $f$ accepts.
We are told $f$ has domain $[0,7]$. That means $f$ only accepts inputs between $0$ and $7$. So we need \[ 0\le g(x)\le 7. \]
Step 3: Put in $g(x)$.
Here $g(x)=|2x+1|$. So the rule becomes \[ 0\le |2x+1|\le 7. \] The left part $0\le|2x+1|$ is always true because an absolute value is never negative.
Step 4: Solve the absolute value inequality.
$|2x+1|\le 7$ means $2x+1$ lies between $-7$ and $7$: \[ -7\le 2x+1\le 7. \]
Step 5: Isolate $x$.
Subtract $1$ from each part: \[ -8\le 2x\le 6. \] Now divide each part by $2$: \[ -4\le x\le 3. \]
Step 6: Write the domain.
So the allowed $x$ values form the closed interval from $-4$ to $3$. \[ \boxed{[-4,3]} \]