QuestionFind $g(f(x))$ for $f(x)=x-3$ and $g(x)=x^{2}-4$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=x-3$ . The function $g(x)$ is defined as $g(x)=x^{}-4$
3. We need to find the composition of the functions $g(f(x))$

STEP 2

The composition of functions is a concept in mathematics where the output of one function becomes the input of another function. In this case, we want to find $g(f(x))$ , which means that the output of $f(x)$ will be the input for $g(x)$ .

STEP 3

First, let's find the output of $f(x)$ .
$f(x) = x -3$

STEP 4

Now, we will use this output as the input for $g(x)$ . We replace every $x$ in $g(x)$ with the output from $f(x)$ , which is $x -3$ .
$g(f(x)) = (f(x))^{2} -4$

STEP 5

Substitute $f(x)$ into the equation.
$g(f(x)) = (x -3)^{2} -4$

STEP 6

Expand the squared term.
$g(f(x)) = x^{2} -6x +9 -4$

STEP 7

implify the equation.
$g(f(x)) = x^{2} -6x +5$ So, $g(f(x)) = x^{2} -6x +5$ .

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QuestionFind $g(f(x))$ for $f(x)=x-3$ and $g(x)=x^{2}-4$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=x-3$ . The function $g(x)$ is defined as $g(x)=x^{}-4$
3. We need to find the composition of the functions $g(f(x))$

STEP 2

The composition of functions is a concept in mathematics where the output of one function becomes the input of another function. In this case, we want to find $g(f(x))$ , which means that the output of $f(x)$ will be the input for $g(x)$ .

STEP 3

First, let's find the output of $f(x)$ .
$f(x) = x -3$

STEP 4

Now, we will use this output as the input for $g(x)$ . We replace every $x$ in $g(x)$ with the output from $f(x)$ , which is $x -3$ .
$g(f(x)) = (f(x))^{2} -4$

STEP 5

Substitute $f(x)$ into the equation.
$g(f(x)) = (x -3)^{2} -4$

STEP 6

Expand the squared term.
$g(f(x)) = x^{2} -6x +9 -4$

STEP 7

implify the equation.
$g(f(x)) = x^{2} -6x +5$ So, $g(f(x)) = x^{2} -6x +5$ .

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QuestionFind g(f(x))g(f(x))g(f(x)) for f(x)=x−3f(x)=x-3f(x)=x−3 and g(x)=x2−4g(x)=x^{2}-4g(x)=x2−4.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x)f(x) is defined as f(x)=x−3f(x)=x-3f(x)=x−3 . The function g(x)g(x)g(x) is defined as g(x)=x−4g(x)=x^{}-4g(x)=x−43. We need to find the composition of the functions g(f(x))g(f(x))g(f(x))

STEP 2

The composition of functions is a concept in mathematics where the output of one function becomes the input of another function. In this case, we want to find g(f(x))g(f(x))g(f(x)), which means that the output of f(x)f(x)f(x) will be the input for g(x)g(x)g(x).

STEP 3

First, let's find the output of f(x)f(x)f(x).f(x)=x−3f(x) = x -3f(x)=x−3

STEP 4

Now, we will use this output as the input for g(x)g(x)g(x). We replace every xxx in g(x)g(x)g(x) with the output from f(x)f(x)f(x), which is x−3x -3x−3.g(f(x))=(f(x))2−4g(f(x)) = (f(x))^{2} -4g(f(x))=(f(x))2−4

STEP 5

Substitute f(x)f(x)f(x) into the equation.g(f(x))=(x−3)2−4g(f(x)) = (x -3)^{2} -4g(f(x))=(x−3)2−4

STEP 6

Expand the squared term.g(f(x))=x2−6x+9−4g(f(x)) = x^{2} -6x +9 -4g(f(x))=x2−6x+9−4

STEP 7

implify the equation.g(f(x))=x2−6x+5g(f(x)) = x^{2} -6x +5g(f(x))=x2−6x+5So, g(f(x))=x2−6x+5g(f(x)) = x^{2} -6x +5g(f(x))=x2−6x+5.

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QuestionFind $g(f(x))$ for $f(x)=x-3$ and $g(x)=x^{2}-4$ .

Assumptions1. The function $f(x)$ is defined as $f(x)=x-3$ . The function $g(x)$ is defined as $g(x)=x^{}-4$
3. We need to find the composition of the functions $g(f(x))$

The composition of functions is a concept in mathematics where the output of one function becomes the input of another function. In this case, we want to find $g(f(x))$ , which means that the output of $f(x)$ will be the input for $g(x)$ .

First, let's find the output of $f(x)$ .
$f(x) = x -3$

Now, we will use this output as the input for $g(x)$ . We replace every $x$ in $g(x)$ with the output from $f(x)$ , which is $x -3$ .
$g(f(x)) = (f(x))^{2} -4$

Substitute $f(x)$ into the equation.
$g(f(x)) = (x -3)^{2} -4$

Expand the squared term.
$g(f(x)) = x^{2} -6x +9 -4$

implify the equation.
$g(f(x)) = x^{2} -6x +5$ So, $g(f(x)) = x^{2} -6x +5$ .