Solved on Feb 06, 2024

Find the value of $h(6)$ given $f(x)=x-5$ , $g(x)=-3x$ , and $h(x)=2f(x+3)+3g(x-3)$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined as $f(x) = x - 5$ .
2. The function $g(x)$ is defined as $g(x) = -3x$ .
3. The function $h(x)$ is defined as $h(x) = 2f(x+3) + 3g(x-3)$ .
4. We need to find the value of $h(6)$ .

STEP 2

First, we need to find the value of $f(x+3)$ . To do this, we substitute $x+3$ into the function $f(x)$ .
$f(x+3) = (x+3) - 5$

STEP 3

Now, we will calculate $f(x+3)$ by substituting $x = 6$ .
$f(6+3) = (6+3) - 5$

STEP 4

Simplify the expression for $f(6+3)$ .
$f(6+3) = 9 - 5$

STEP 5

Calculate the value of $f(6+3)$ .
$f(6+3) = 4$

STEP 6

Next, we need to find the value of $g(x-3)$ . To do this, we substitute $x-3$ into the function $g(x)$ .
$g(x-3) = -3(x-3)$

STEP 7

Now, we will calculate $g(x-3)$ by substituting $x = 6$ .
$g(6-3) = -3(6-3)$

STEP 8

Simplify the expression for $g(6-3)$ .
$g(6-3) = -3 \cdot 3$

STEP 9

Calculate the value of $g(6-3)$ .
$g(6-3) = -9$

STEP 10

Now that we have the values of $f(x+3)$ and $g(x-3)$ , we can find the value of $h(x)$ by substituting these values into the definition of $h(x)$ .
$h(x) = 2f(x+3) + 3g(x-3)$

STEP 11

Substitute $f(6+3)$ and $g(6-3)$ into the expression for $h(x)$ .
$h(6) = 2f(6+3) + 3g(6-3)$

STEP 12

Substitute the calculated values from STEP_5 and STEP_9.
$h(6) = 2 \cdot 4 + 3 \cdot (-9)$

STEP 13

Simplify the expression for $h(6)$ .
$h(6) = 8 - 27$

STEP 14

Calculate the value of $h(6)$ .
$h(6) = -19$
The value of $h(6)$ is $-19$ .

Was this helpful?

Find the value of $h(6)$ given $f(x)=x-5$ , $g(x)=-3x$ , and $h(x)=2f(x+3)+3g(x-3)$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined as $f(x) = x - 5$ .
2. The function $g(x)$ is defined as $g(x) = -3x$ .
3. The function $h(x)$ is defined as $h(x) = 2f(x+3) + 3g(x-3)$ .
4. We need to find the value of $h(6)$ .

STEP 2

First, we need to find the value of $f(x+3)$ . To do this, we substitute $x+3$ into the function $f(x)$ .
$f(x+3) = (x+3) - 5$

STEP 3

Now, we will calculate $f(x+3)$ by substituting $x = 6$ .
$f(6+3) = (6+3) - 5$

STEP 4

Simplify the expression for $f(6+3)$ .
$f(6+3) = 9 - 5$

STEP 5

Calculate the value of $f(6+3)$ .
$f(6+3) = 4$

STEP 6

Next, we need to find the value of $g(x-3)$ . To do this, we substitute $x-3$ into the function $g(x)$ .
$g(x-3) = -3(x-3)$

STEP 7

Now, we will calculate $g(x-3)$ by substituting $x = 6$ .
$g(6-3) = -3(6-3)$

STEP 8

Simplify the expression for $g(6-3)$ .
$g(6-3) = -3 \cdot 3$

STEP 9

Calculate the value of $g(6-3)$ .
$g(6-3) = -9$

STEP 10

Now that we have the values of $f(x+3)$ and $g(x-3)$ , we can find the value of $h(x)$ by substituting these values into the definition of $h(x)$ .
$h(x) = 2f(x+3) + 3g(x-3)$

STEP 11

Substitute $f(6+3)$ and $g(6-3)$ into the expression for $h(x)$ .
$h(6) = 2f(6+3) + 3g(6-3)$

STEP 12

Substitute the calculated values from STEP_5 and STEP_9.
$h(6) = 2 \cdot 4 + 3 \cdot (-9)$

STEP 13

Simplify the expression for $h(6)$ .
$h(6) = 8 - 27$

STEP 14

Calculate the value of $h(6)$ .
$h(6) = -19$
The value of $h(6)$ is $-19$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the value of h(6)h(6)h(6) given f(x)=x−5f(x)=x-5f(x)=x−5, g(x)=−3xg(x)=-3xg(x)=−3x, and h(x)=2f(x+3)+3g(x−3)h(x)=2f(x+3)+3g(x-3)h(x)=2f(x+3)+3g(x−3).

STEP 1

STEP 2

First, we need to find the value of f(x+3)f(x+3)f(x+3). To do this, we substitute x+3x+3x+3 into the function f(x)f(x)f(x).f(x+3)=(x+3)−5f(x+3) = (x+3) - 5f(x+3)=(x+3)−5

STEP 3

Now, we will calculate f(x+3)f(x+3)f(x+3) by substituting x=6x = 6x=6.f(6+3)=(6+3)−5f(6+3) = (6+3) - 5f(6+3)=(6+3)−5

STEP 4

Simplify the expression for f(6+3)f(6+3)f(6+3).f(6+3)=9−5f(6+3) = 9 - 5f(6+3)=9−5

STEP 5

Calculate the value of f(6+3)f(6+3)f(6+3).f(6+3)=4f(6+3) = 4f(6+3)=4

STEP 6

Next, we need to find the value of g(x−3)g(x-3)g(x−3). To do this, we substitute x−3x-3x−3 into the function g(x)g(x)g(x).g(x−3)=−3(x−3)g(x-3) = -3(x-3)g(x−3)=−3(x−3)

STEP 7

Now, we will calculate g(x−3)g(x-3)g(x−3) by substituting x=6x = 6x=6.g(6−3)=−3(6−3)g(6-3) = -3(6-3)g(6−3)=−3(6−3)

STEP 8

Simplify the expression for g(6−3)g(6-3)g(6−3).g(6−3)=−3⋅3g(6-3) = -3 \cdot 3g(6−3)=−3⋅3

STEP 9

Calculate the value of g(6−3)g(6-3)g(6−3).g(6−3)=−9g(6-3) = -9g(6−3)=−9

STEP 10

Now that we have the values of f(x+3)f(x+3)f(x+3) and g(x−3)g(x-3)g(x−3), we can find the value of h(x)h(x)h(x) by substituting these values into the definition of h(x)h(x)h(x).h(x)=2f(x+3)+3g(x−3)h(x) = 2f(x+3) + 3g(x-3)h(x)=2f(x+3)+3g(x−3)

STEP 11

Substitute f(6+3)f(6+3)f(6+3) and g(6−3)g(6-3)g(6−3) into the expression for h(x)h(x)h(x).h(6)=2f(6+3)+3g(6−3)h(6) = 2f(6+3) + 3g(6-3)h(6)=2f(6+3)+3g(6−3)

STEP 12

Substitute the calculated values from STEP_5 and STEP_9.h(6)=2⋅4+3⋅(−9)h(6) = 2 \cdot 4 + 3 \cdot (-9)h(6)=2⋅4+3⋅(−9)

STEP 13

Simplify the expression for h(6)h(6)h(6).h(6)=8−27h(6) = 8 - 27h(6)=8−27

STEP 14

Calculate the value of h(6)h(6)h(6).h(6)=−19h(6) = -19h(6)=−19The value of h(6)h(6)h(6) is −19-19−19.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the value of $h(6)$ given $f(x)=x-5$ , $g(x)=-3x$ , and $h(x)=2f(x+3)+3g(x-3)$ .

First, we need to find the value of $f(x+3)$ . To do this, we substitute $x+3$ into the function $f(x)$ .
$f(x+3) = (x+3) - 5$

Now, we will calculate $f(x+3)$ by substituting $x = 6$ .
$f(6+3) = (6+3) - 5$

Simplify the expression for $f(6+3)$ .
$f(6+3) = 9 - 5$

Calculate the value of $f(6+3)$ .
$f(6+3) = 4$

Next, we need to find the value of $g(x-3)$ . To do this, we substitute $x-3$ into the function $g(x)$ .
$g(x-3) = -3(x-3)$

Now, we will calculate $g(x-3)$ by substituting $x = 6$ .
$g(6-3) = -3(6-3)$

Simplify the expression for $g(6-3)$ .
$g(6-3) = -3 \cdot 3$

Calculate the value of $g(6-3)$ .
$g(6-3) = -9$

Now that we have the values of $f(x+3)$ and $g(x-3)$ , we can find the value of $h(x)$ by substituting these values into the definition of $h(x)$ .
$h(x) = 2f(x+3) + 3g(x-3)$

Substitute $f(6+3)$ and $g(6-3)$ into the expression for $h(x)$ .
$h(6) = 2f(6+3) + 3g(6-3)$

Substitute the calculated values from STEP_5 and STEP_9.
$h(6) = 2 \cdot 4 + 3 \cdot (-9)$

Simplify the expression for $h(6)$ .
$h(6) = 8 - 27$

Calculate the value of $h(6)$ .
$h(6) = -19$
The value of $h(6)$ is $-19$ .