Solved on Dec 12, 2023

Find the composite function of two given linear functions. Given $f(x) = 4x + 14$ and $g(x) = 2x - 1$ , find $(f \circ g)(x)$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined as $f(x) = 4x + 14$ .
2. The function $g(x)$ is defined as $g(x) = 2x - 1$ .
3. The composition of the two functions $(f \circ g)(x)$ means $f(g(x))$ .

STEP 2

To find the composition $(f \circ g)(x)$ , we need to substitute $g(x)$ into the function $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Now, we substitute $g(x) = 2x - 1$ into $f(x)$ .
$(f \circ g)(x) = f(2x - 1)$

STEP 4

Replace $x$ in $f(x) = 4x + 14$ with $(2x - 1)$ .
$(f \circ g)(x) = 4(2x - 1) + 14$

STEP 5

Distribute the $4$ across the terms inside the parentheses.
$(f \circ g)(x) = 4 \cdot 2x - 4 \cdot 1 + 14$

STEP 6

Perform the multiplication.
$(f \circ g)(x) = 8x - 4 + 14$

STEP 7

Combine the constant terms $-4$ and $14$ .
$(f \circ g)(x) = 8x + 10$
Thus, the composition of the functions $(f \circ g)(x)$ is $8x + 10$ .

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Find the composite function of two given linear functions. Given $f(x) = 4x + 14$ and $g(x) = 2x - 1$ , find $(f \circ g)(x)$ .

STEP 1

Assumptions
1. The function $f(x)$ is defined as $f(x) = 4x + 14$ .
2. The function $g(x)$ is defined as $g(x) = 2x - 1$ .
3. The composition of the two functions $(f \circ g)(x)$ means $f(g(x))$ .

STEP 2

To find the composition $(f \circ g)(x)$ , we need to substitute $g(x)$ into the function $f(x)$ .
$(f \circ g)(x) = f(g(x))$

STEP 3

Now, we substitute $g(x) = 2x - 1$ into $f(x)$ .
$(f \circ g)(x) = f(2x - 1)$

STEP 4

Replace $x$ in $f(x) = 4x + 14$ with $(2x - 1)$ .
$(f \circ g)(x) = 4(2x - 1) + 14$

STEP 5

Distribute the $4$ across the terms inside the parentheses.
$(f \circ g)(x) = 4 \cdot 2x - 4 \cdot 1 + 14$

STEP 6

Perform the multiplication.
$(f \circ g)(x) = 8x - 4 + 14$

STEP 7

Combine the constant terms $-4$ and $14$ .
$(f \circ g)(x) = 8x + 10$
Thus, the composition of the functions $(f \circ g)(x)$ is $8x + 10$ .

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Find the composite function of two given linear functions. Given f(x)=4x+14f(x) = 4x + 14f(x)=4x+14 and g(x)=2x−1g(x) = 2x - 1g(x)=2x−1, find (f∘g)(x)(f \circ g)(x)(f∘g)(x).

STEP 1

Assumptions1. The function f(x)f(x)f(x) is defined as f(x)=4x+14f(x) = 4x + 14f(x)=4x+14.2. The function g(x)g(x)g(x) is defined as g(x)=2x−1g(x) = 2x - 1g(x)=2x−1.3. The composition of the two functions (f∘g)(x)(f \circ g)(x)(f∘g)(x) means f(g(x))f(g(x))f(g(x)).

STEP 2

To find the composition (f∘g)(x)(f \circ g)(x)(f∘g)(x), we need to substitute g(x)g(x)g(x) into the function f(x)f(x)f(x).(f∘g)(x)=f(g(x)) (f \circ g)(x) = f(g(x)) (f∘g)(x)=f(g(x))

STEP 3

Now, we substitute g(x)=2x−1g(x) = 2x - 1g(x)=2x−1 into f(x)f(x)f(x).(f∘g)(x)=f(2x−1) (f \circ g)(x) = f(2x - 1) (f∘g)(x)=f(2x−1)

STEP 4

Replace xxx in f(x)=4x+14f(x) = 4x + 14f(x)=4x+14 with (2x−1)(2x - 1)(2x−1).(f∘g)(x)=4(2x−1)+14 (f \circ g)(x) = 4(2x - 1) + 14 (f∘g)(x)=4(2x−1)+14

STEP 5

Distribute the 444 across the terms inside the parentheses.(f∘g)(x)=4⋅2x−4⋅1+14 (f \circ g)(x) = 4 \cdot 2x - 4 \cdot 1 + 14 (f∘g)(x)=4⋅2x−4⋅1+14

STEP 6

Perform the multiplication.(f∘g)(x)=8x−4+14 (f \circ g)(x) = 8x - 4 + 14 (f∘g)(x)=8x−4+14

STEP 7

Combine the constant terms −4-4−4 and 141414.(f∘g)(x)=8x+10 (f \circ g)(x) = 8x + 10 (f∘g)(x)=8x+10Thus, the composition of the functions (f∘g)(x)(f \circ g)(x)(f∘g)(x) is 8x+108x + 108x+10.

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Find the composite function of two given linear functions. Given $f(x) = 4x + 14$ and $g(x) = 2x - 1$ , find $(f \circ g)(x)$ .

Assumptions
1. The function $f(x)$ is defined as $f(x) = 4x + 14$ .
2. The function $g(x)$ is defined as $g(x) = 2x - 1$ .
3. The composition of the two functions $(f \circ g)(x)$ means $f(g(x))$ .

To find the composition $(f \circ g)(x)$ , we need to substitute $g(x)$ into the function $f(x)$ .
$(f \circ g)(x) = f(g(x))$

Now, we substitute $g(x) = 2x - 1$ into $f(x)$ .
$(f \circ g)(x) = f(2x - 1)$

Replace $x$ in $f(x) = 4x + 14$ with $(2x - 1)$ .
$(f \circ g)(x) = 4(2x - 1) + 14$

Distribute the $4$ across the terms inside the parentheses.
$(f \circ g)(x) = 4 \cdot 2x - 4 \cdot 1 + 14$

Perform the multiplication.
$(f \circ g)(x) = 8x - 4 + 14$

Combine the constant terms $-4$ and $14$ .
$(f \circ g)(x) = 8x + 10$
Thus, the composition of the functions $(f \circ g)(x)$ is $8x + 10$ .