Question Evaluate the composition of $f(x)=x^2-13$ and $g(x)=5-x$ at $x=-3$ .

Studdy Solution

STEP 1

1. The composition of functions $(fg)(x)$ means applying the function $g$ to $x$ first and then applying the function $f$ to the result of $g(x)$ .
2. The functions $f(x)$ and $g(x)$ are defined for all real numbers.
3. The composition $(fg)(-3)$ means we first find $g(-3)$ and then apply $f$ to that result.

STEP 2

1. Evaluate $g(-3)$ .
2. Substitute the result from step 1 into $f(x)$ .
3. Simplify to find the final result.

STEP 3

Calculate $g(-3)$ using the definition of $g(x)$ .
$g(-3) = 5 - (-3)$

STEP 4

Simplify the expression to find the value of $g(-3)$ .
$g(-3) = 5 + 3$

STEP 5

Complete the calculation to get the numerical value.
$g(-3) = 8$

STEP 6

Substitute the result from STEP_3 into $f(x)$ to get $f(g(-3))$ .
$f(g(-3)) = f(8)$

STEP 7

Evaluate $f(8)$ using the definition of $f(x)$ .
$f(8) = 8^2 - 13$

STEP 8

Calculate the square of 8 and subtract 13 to find the value of $f(8)$ .
$f(8) = 64 - 13$

STEP 9

Complete the calculation to get the numerical value.
$f(8) = 51$

STEP 10

Conclude that $(fg)(-3)$ is equal to the value found in STEP_7.
$(fg)(-3) = 51$

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