Math / Algebra

QuestionFind functions $f$ and $g$ so that $f \circ g=H$ . $H(x)=(5 x+2)^{4}$
Choose the correct pair of functions. A. $f(x)=\sqrt[4]{x}, g(x)=\frac{x-2}{5}$ C. $f(x)=\frac{x-2}{5}, g(x)=\sqrt[4]{x}$ $B$ . $f(x)=x^{4}, g(x)=5 x+2$ D. $f(x)=5 x+2, g(x)=x^{4}$

Studdy Solution

STEP 1

1. We are given the composite function $H(x) = (5x + 2)^4$ .
2. We need to find functions $f$ and $g$ such that $f \circ g = H$ .

STEP 2

1. Understand the structure of the function $H(x)$ .
2. Analyze each option to determine if $f \circ g = H$ .
3. Verify the correct pair of functions.

STEP 3

Understand the structure of $H(x) = (5x + 2)^4$ .
- The function $H(x)$ is a composition of two operations: first, a linear transformation $5x + 2$ , and second, raising the result to the 4th power.

STEP 4

Analyze Option A:
- $f(x) = \sqrt[4]{x}$ , $g(x) = \frac{x-2}{5}$ - Compute $f(g(x)) = f\left(\frac{x-2}{5}\right) = \sqrt[4]{\frac{x-2}{5}}$
This does not match $H(x) = (5x + 2)^4$ .

STEP 5

Analyze Option B:
- $f(x) = x^4$ , $g(x) = 5x + 2$ - Compute $f(g(x)) = f(5x + 2) = (5x + 2)^4$
This matches $H(x) = (5x + 2)^4$ .

STEP 6

Analyze Option C:
- $f(x) = \frac{x-2}{5}$ , $g(x) = \sqrt[4]{x}$ - Compute $f(g(x)) = f(\sqrt[4]{x}) = \frac{\sqrt[4]{x} - 2}{5}$
This does not match $H(x) = (5x + 2)^4$ .

STEP 7

Analyze Option D:
- $f(x) = 5x + 2$ , $g(x) = x^4$ - Compute $f(g(x)) = f(x^4) = 5(x^4) + 2$
This does not match $H(x) = (5x + 2)^4$ .

STEP 8

Verify the correct pair of functions:
- From the analysis, Option B is the correct pair: $f(x) = x^4$ , $g(x) = 5x + 2$ .
The correct pair of functions is:
$\boxed{B}$

Was this helpful?

QuestionFind functions $f$ and $g$ so that $f \circ g=H$ . $H(x)=(5 x+2)^{4}$
Choose the correct pair of functions. A. $f(x)=\sqrt[4]{x}, g(x)=\frac{x-2}{5}$ C. $f(x)=\frac{x-2}{5}, g(x)=\sqrt[4]{x}$ $B$ . $f(x)=x^{4}, g(x)=5 x+2$ D. $f(x)=5 x+2, g(x)=x^{4}$

Studdy Solution

STEP 1

1. We are given the composite function $H(x) = (5x + 2)^4$ .
2. We need to find functions $f$ and $g$ such that $f \circ g = H$ .

STEP 2

1. Understand the structure of the function $H(x)$ .
2. Analyze each option to determine if $f \circ g = H$ .
3. Verify the correct pair of functions.

STEP 3

Understand the structure of $H(x) = (5x + 2)^4$ .
- The function $H(x)$ is a composition of two operations: first, a linear transformation $5x + 2$ , and second, raising the result to the 4th power.

STEP 4

Analyze Option A:
- $f(x) = \sqrt[4]{x}$ , $g(x) = \frac{x-2}{5}$ - Compute $f(g(x)) = f\left(\frac{x-2}{5}\right) = \sqrt[4]{\frac{x-2}{5}}$
This does not match $H(x) = (5x + 2)^4$ .

STEP 5

Analyze Option B:
- $f(x) = x^4$ , $g(x) = 5x + 2$ - Compute $f(g(x)) = f(5x + 2) = (5x + 2)^4$
This matches $H(x) = (5x + 2)^4$ .

STEP 6

Analyze Option C:
- $f(x) = \frac{x-2}{5}$ , $g(x) = \sqrt[4]{x}$ - Compute $f(g(x)) = f(\sqrt[4]{x}) = \frac{\sqrt[4]{x} - 2}{5}$
This does not match $H(x) = (5x + 2)^4$ .

STEP 7

Analyze Option D:
- $f(x) = 5x + 2$ , $g(x) = x^4$ - Compute $f(g(x)) = f(x^4) = 5(x^4) + 2$
This does not match $H(x) = (5x + 2)^4$ .

STEP 8

Verify the correct pair of functions:
- From the analysis, Option B is the correct pair: $f(x) = x^4$ , $g(x) = 5x + 2$ .
The correct pair of functions is:
$\boxed{B}$

Get Studdy

Studdy solves anything!

Start learning now

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

Studdy Solution

STEP 1

1. We are given the composite function H(x)=(5x+2)4 H(x) = (5x + 2)^4 H(x)=(5x+2)4.2. We need to find functions f f f and g g g such that f∘g=H f \circ g = H f∘g=H.

STEP 2

1. Understand the structure of the function H(x) H(x) H(x).2. Analyze each option to determine if f∘g=H f \circ g = H f∘g=H.3. Verify the correct pair of functions.

STEP 3

Understand the structure of H(x)=(5x+2)4 H(x) = (5x + 2)^4 H(x)=(5x+2)4.- The function H(x) H(x) H(x) is a composition of two operations: first, a linear transformation 5x+2 5x + 2 5x+2, and second, raising the result to the 4th power.

STEP 4

STEP 5

Analyze Option B:- f(x)=x4 f(x) = x^4 f(x)=x4, g(x)=5x+2 g(x) = 5x + 2 g(x)=5x+2 - Compute f(g(x))=f(5x+2)=(5x+2)4 f(g(x)) = f(5x + 2) = (5x + 2)^4 f(g(x))=f(5x+2)=(5x+2)4This matches H(x)=(5x+2)4 H(x) = (5x + 2)^4 H(x)=(5x+2)4.

STEP 6

Analyze Option C:- f(x)=x−25 f(x) = \frac{x-2}{5} f(x)=5x−2​, g(x)=x4 g(x) = \sqrt[4]{x} g(x)=4x​ - Compute f(g(x))=f(x4)=x4−25 f(g(x)) = f(\sqrt[4]{x}) = \frac{\sqrt[4]{x} - 2}{5} f(g(x))=f(4x​)=54x​−2​This does not match H(x)=(5x+2)4 H(x) = (5x + 2)^4 H(x)=(5x+2)4.

STEP 7

Analyze Option D:- f(x)=5x+2 f(x) = 5x + 2 f(x)=5x+2, g(x)=x4 g(x) = x^4 g(x)=x4 - Compute f(g(x))=f(x4)=5(x4)+2 f(g(x)) = f(x^4) = 5(x^4) + 2 f(g(x))=f(x4)=5(x4)+2This does not match H(x)=(5x+2)4 H(x) = (5x + 2)^4 H(x)=(5x+2)4.

STEP 8

Verify the correct pair of functions:- From the analysis, Option B is the correct pair: f(x)=x4 f(x) = x^4 f(x)=x4, g(x)=5x+2 g(x) = 5x + 2 g(x)=5x+2. The correct pair of functions is:B \boxed{B} B​

Get Studdy

Studdy solves anything!

Start learning now

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

1. We are given the composite function $H(x) = (5x + 2)^4$ .
2. We need to find functions $f$ and $g$ such that $f \circ g = H$ .

1. Understand the structure of the function $H(x)$ .
2. Analyze each option to determine if $f \circ g = H$ .
3. Verify the correct pair of functions.

Understand the structure of $H(x) = (5x + 2)^4$ .
- The function $H(x)$ is a composition of two operations: first, a linear transformation $5x + 2$ , and second, raising the result to the 4th power.

Analyze Option B:
- $f(x) = x^4$ , $g(x) = 5x + 2$ - Compute $f(g(x)) = f(5x + 2) = (5x + 2)^4$
This matches $H(x) = (5x + 2)^4$ .

Analyze Option C:
- $f(x) = \frac{x-2}{5}$ , $g(x) = \sqrt[4]{x}$ - Compute $f(g(x)) = f(\sqrt[4]{x}) = \frac{\sqrt[4]{x} - 2}{5}$
This does not match $H(x) = (5x + 2)^4$ .

Analyze Option D:
- $f(x) = 5x + 2$ , $g(x) = x^4$ - Compute $f(g(x)) = f(x^4) = 5(x^4) + 2$
This does not match $H(x) = (5x + 2)^4$ .

Verify the correct pair of functions:
- From the analysis, Option B is the correct pair: $f(x) = x^4$ , $g(x) = 5x + 2$ .
The correct pair of functions is:
$\boxed{B}$