QuestionGiven that $f(x)=x^{2}+5 x$ and $g(x)=x-6$ , calculate the following: You do not need to simplify (a) $(f \circ g)(x)=$ $\square$ (b) $(g \circ f)(x)=$ $\square$ (c) $(f \circ f)(x)=$ $\square$ (d) $(g \circ g)(x)=$ $\square$

Studdy Solution

STEP 1

1. The notation $(f \circ g)(x)$ represents the composition of the functions $f$ and $g$ , meaning $f(g(x))$ .
2. The function $f(x) = x^2 + 5x$ is a quadratic function.
3. The function $g(x) = x - 6$ is a linear function.
4. We will substitute the expression for one function into the other as indicated by the composition.

STEP 2

1. Calculate $(f \circ g)(x)$ .
2. Calculate $(g \circ f)(x)$ .
3. Calculate $(f \circ f)(x)$ .
4. Calculate $(g \circ g)(x)$ .

STEP 3

To find $(f \circ g)(x)$ , substitute $g(x)$ into $f(x)$ :
$f(g(x)) = f(x - 6)$
Substitute $x - 6$ into $f(x) = x^2 + 5x$ :
$f(x - 6) = (x - 6)^2 + 5(x - 6)$

STEP 4

To find $(g \circ f)(x)$ , substitute $f(x)$ into $g(x)$ :
$g(f(x)) = g(x^2 + 5x)$
Substitute $x^2 + 5x$ into $g(x) = x - 6$ :
$g(x^2 + 5x) = (x^2 + 5x) - 6$

STEP 5

To find $(f \circ f)(x)$ , substitute $f(x)$ into itself:
$f(f(x)) = f(x^2 + 5x)$
Substitute $x^2 + 5x$ into $f(x) = x^2 + 5x$ :
$f(x^2 + 5x) = (x^2 + 5x)^2 + 5(x^2 + 5x)$

STEP 6

To find $(g \circ g)(x)$ , substitute $g(x)$ into itself:
$g(g(x)) = g(x - 6)$
Substitute $x - 6$ into $g(x) = x - 6$ :
$g(x - 6) = (x - 6) - 6$
The solutions are: (a) $(f \circ g)(x) = (x - 6)^2 + 5(x - 6)$ (b) $(g \circ f)(x) = x^2 + 5x - 6$ (c) $(f \circ f)(x) = (x^2 + 5x)^2 + 5(x^2 + 5x)$ (d) $(g \circ g)(x) = x - 12$

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