QuestionFind the domain of the composite function $f(g(x))$ where $f(x)=\sqrt{42-x}$ and $g(x)=x^{2}-x$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)=\sqrt{42-x}$ is a square root function and its domain is all real numbers $x$ such that $42-x \geq0$ . . The function $g(x)=x^{}-x$ is a quadratic function and its domain is all real numbers.
3. The composition of functions $f \circ g$ means $f(g(x))$ .

STEP 2

First, we need to find the domain of the function $f(x)$ .
The domain of $f(x)$ is all real numbers $x$ such that $42-x \geq0$ .

STEP 3

olving the inequality $42-x \geq0$ gives us the domain of $f(x)$ .
$42-x \geq0$ $x \leq42$ So, the domain of $f(x)$ is all real numbers $x$ such that $x \leq42$ .

STEP 4

Now, we need to find the domain of the function $g(x)$ .
The domain of $g(x)$ is all real numbers.

STEP 5

The composition of functions $f \circ g$ means $f(g(x))$ .So, we need to find the domain of the function $f(g(x))$ .

STEP 6

To find the domain of $f(g(x))$ , we need to ensure that the output of $g(x)$ is within the domain of $f(x)$ .
So, we need to find all real numbers $x$ such that $g(x) \leq42$ .

STEP 7

Substitute $g(x) = x^{2}-x$ into the inequality $g(x) \leq42$ .
$x^{2}-x \leq42$

STEP 8

Rearrange the inequality to make it a quadratic inequality.
$x^{2}-x-42 \leq0$

STEP 9

Factor the quadratic inequality.
$(x-7)(x+6) \leq$

STEP 10

olve the inequality $(x-7)(x+6) \leq0$ to find the domain of $f(g(x))$ .
The solutions of the inequality are $x=7$ and $x=-6$ .

STEP 11

The inequality $(x-7)(x+6) \leq0$ is satisfied when $-6 \leq x \leq7$ .
So, the domain of $f \circ g$ is all real numbers $x$ such that $-6 \leq x \leq7$ .

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