QuestionFind the value not in the range of $f(g(x))$ where $f(x)=5-2x$ and $g(x)=\frac{x^{2}}{4}$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x) =5 -x$ . The function $g(x)$ is defined as $g(x) = \frac{x^{}}{4}$
3. We need to find the range of the composite function $f(g(x))$

STEP 2

First, we need to find the composite function $f(g(x))$ . This is done by substituting $g(x)$ into $f(x)$ .
$f(g(x)) = f\left(\frac{x^{2}}{4}\right)$

STEP 3

Substitute $\frac{x^{2}}{}$ for $x$ in $f(x)$ to get $f(g(x))$ .
$f(g(x)) =5 -2\left(\frac{x^{2}}{}\right)$

STEP 4

implify the expression to get the composite function $f(g(x))$ .
$f(g(x)) = - \frac{x^{2}}{2}$

STEP 5

Now, we need to find the range of $f(g(x))$ . The range of a function is the set of all possible output values (y-values), which are determined by the domain of the function.For the function $f(g(x)) =5 - \frac{x^{2}}{2}$ , the domain is all real numbers because $x$ can be any real number.

STEP 6

The function $f(g(x)) =5 - \frac{x^{2}}{2}$ is a downward-opening parabola with a vertex at $(0,5)$ . This means that the maximum value of the function is5, and it can take on any value less than or equal to5.Therefore, the range of $f(g(x))$ is $(-\infty,5]$ .

STEP 7

Any number greater than5 is not in the range of $f(g(x))$ .

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