QuestionEvaluate $(q \circ p)(x)$ for $p(x)=x^{2}-9 x$ and $q(x)=\frac{1}{x-10}$ . Find its domain in interval notation.

Studdy Solution

STEP 1

Assumptions1. The function $p(x)$ is defined as $x^{}-9x$ . The function $q(x)$ is defined as $\frac{1}{x-10}$
3. We need to find the composite function $(q \circ p)(x)$ and its domain

STEP 2

First, we need to find the composite function $(q \circ p)(x)$ . This is done by substituting $p(x)$ into $q(x)$ .
$(q \circ p)(x) = q(p(x))$

STEP 3

Now, plug in the given functions $p(x)$ and $q(x)$ to calculate the composite function.
$(q \circ p)(x) = q(x^{2}-9x)$

STEP 4

Substitute $p(x)$ into $q(x)$ .
$(q \circ p)(x) = \frac{1}{x^{2}-9x-10}$

STEP 5

Now, we need to find the domain of the composite function $(q \circ p)(x)$ . The domain of a function is the set of all real numbers for which the function is defined. For the function $q(x) = \frac{1}{x-10}$ , the function is undefined when $x-10 =0$ , or $x =10$ .

STEP 6

Similarly, for the function $(q \circ p)(x) = \frac{1}{x^{2}-9x-10}$ , the function is undefined when $x^{2}-9x-10 =0$ . We can find the values of $x$ that make this equation true by factoring the quadratic equation.

STEP 7

Factor the quadratic equation $x^{2}-9x-10 =0$ .
$(x -10)(x +1) =0$

STEP 8

Set each factor equal to zero and solve for $x$ .
$x -10 =0 \quad \text{or} \quad x +1 =0$

STEP 9

olve each equation for $x$ .
$x = \quad \text{or} \quad x = -$

STEP 10

The domain of the composite function $(q \circ p)(x)$ is all real numbers except $x =10$ and $x = -$ . In interval notation, this is written as $(-\infty, -) \cup (-,10) \cup (10, \infty)$

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