Questions: Suppose f(x)=4-x^2 and g(x)=2x+5. Find the value of f(g(-2)).

Transcript text: Suppose $f(x)=4-x^{2}$ and $g(x)=2 x+5$. Find the value of $f(g(-2))$. 5 $-8$ 3 0

Solution

Step 1: Define the Functions

We are given two functions: \[ f(x) = 4 - x^{2} \] \[ g(x) = 2x + 5 \]

Step 2: Compute the Composite Function

We need to find the composition $f(g(x))$. This means we will substitute $g(x)$ into $f(x)$: \[ f(g(x)) = f(2x + 5) = 4 - (2x + 5)^{2} \]

Step 3: Simplify the Expression

Expanding $f(g(x))$: \[ (2x + 5)^{2} = 4x^{2} + 20x + 25 \] Thus, \[ f(g(x)) = 4 - (4x^{2} + 20x + 25) = -4x^{2} - 20x - 21 \]

Step 4: Determine the Domain and Range

The domain of $f(g(x))$ is $\mathbb{R}$ (all real numbers), and the range is $\left(-\infty, 4\right]$.

Step 5: Evaluate at Specific Value

Now, we evaluate $f(g(-2))$: \[ g(-2) = 2(-2) + 5 = -4 + 5 = 1 \] Then, \[ f(g(-2)) = f(1) = 4 - 1^{2} = 4 - 1 = 3 \]

The value of $f(g(-2))$ is $\boxed{3}$.

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