Math / Algebra

QuestionIf $(f \circ g)(x)=-4 x \cdot(x+1)$ , then find the functions $f(x$ and $g(x)$ . A) $f(x)=x^{2}-1, g(x)=2 x-1$ B) $f(x)=1-x^{2}, g(x)=2 x+1$ C) $f(x)=1-x, g(x)=2 x^{2}+1$ D) $f(x)=2 x+1, g(x)=x^{2}-1$

Studdy Solution

STEP 1

1. We are given that $(f \circ g)(x) = -4x(x+1)$ .
2. We need to determine the functions $f(x)$ and $g(x)$ from the given options.

STEP 2

1. Understand the composition of functions.
2. Evaluate each option by substituting $g(x)$ into $f(x)$ .
3. Compare the result with the given expression $-4x(x+1)$ .
4. Identify the correct pair of functions.

STEP 3

Understand that $(f \circ g)(x) = f(g(x))$ .

STEP 4

Evaluate Option A:
- $f(x) = x^2 - 1$ - $g(x) = 2x - 1$
Substitute $g(x)$ into $f(x)$ :
$f(g(x)) = (2x - 1)^2 - 1$

STEP 5

Simplify the expression:
$f(g(x)) = (2x - 1)^2 - 1$ $= (4x^2 - 4x + 1) - 1$ $= 4x^2 - 4x$
Compare with $-4x(x+1)$ :
$4x^2 - 4x \neq -4x(x+1)$
Option A is incorrect.

STEP 6

Evaluate Option B:
- $f(x) = 1 - x^2$ - $g(x) = 2x + 1$
Substitute $g(x)$ into $f(x)$ :
$f(g(x)) = 1 - (2x + 1)^2$

STEP 7

Simplify the expression:
$f(g(x)) = 1 - (4x^2 + 4x + 1)$ $= 1 - 4x^2 - 4x - 1$ $= -4x^2 - 4x$
Compare with $-4x(x+1)$ :
$-4x^2 - 4x = -4x(x+1)$
Option B is correct.

STEP 8

Since Option B matches the given expression, we have found the correct functions:
$f(x) = 1 - x^2$ $g(x) = 2x + 1$
The correct pair of functions is:
$\boxed{B: f(x) = 1 - x^2, g(x) = 2x + 1}$

Was this helpful?

QuestionIf $(f \circ g)(x)=-4 x \cdot(x+1)$ , then find the functions $f(x$ and $g(x)$ . A) $f(x)=x^{2}-1, g(x)=2 x-1$ B) $f(x)=1-x^{2}, g(x)=2 x+1$ C) $f(x)=1-x, g(x)=2 x^{2}+1$ D) $f(x)=2 x+1, g(x)=x^{2}-1$

Studdy Solution

STEP 1

1. We are given that $(f \circ g)(x) = -4x(x+1)$ .
2. We need to determine the functions $f(x)$ and $g(x)$ from the given options.

STEP 2

1. Understand the composition of functions.
2. Evaluate each option by substituting $g(x)$ into $f(x)$ .
3. Compare the result with the given expression $-4x(x+1)$ .
4. Identify the correct pair of functions.

STEP 3

Understand that $(f \circ g)(x) = f(g(x))$ .

STEP 4

Evaluate Option A:
- $f(x) = x^2 - 1$ - $g(x) = 2x - 1$
Substitute $g(x)$ into $f(x)$ :
$f(g(x)) = (2x - 1)^2 - 1$

STEP 5

Simplify the expression:
$f(g(x)) = (2x - 1)^2 - 1$ $= (4x^2 - 4x + 1) - 1$ $= 4x^2 - 4x$
Compare with $-4x(x+1)$ :
$4x^2 - 4x \neq -4x(x+1)$
Option A is incorrect.

STEP 6

Evaluate Option B:
- $f(x) = 1 - x^2$ - $g(x) = 2x + 1$
Substitute $g(x)$ into $f(x)$ :
$f(g(x)) = 1 - (2x + 1)^2$

STEP 7

Simplify the expression:
$f(g(x)) = 1 - (4x^2 + 4x + 1)$ $= 1 - 4x^2 - 4x - 1$ $= -4x^2 - 4x$
Compare with $-4x(x+1)$ :
$-4x^2 - 4x = -4x(x+1)$
Option B is correct.

STEP 8

Since Option B matches the given expression, we have found the correct functions:
$f(x) = 1 - x^2$ $g(x) = 2x + 1$
The correct pair of functions is:
$\boxed{B: f(x) = 1 - x^2, g(x) = 2x + 1}$

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 that (f∘g)(x)=−4x(x+1) (f \circ g)(x) = -4x(x+1) (f∘g)(x)=−4x(x+1).2. We need to determine the functions f(x) f(x) f(x) and g(x) g(x) g(x) from the given options.

STEP 2

1. Understand the composition of functions.2. Evaluate each option by substituting g(x) g(x) g(x) into f(x) f(x) f(x).3. Compare the result with the given expression −4x(x+1) -4x(x+1) −4x(x+1).4. Identify the correct pair of functions.

STEP 3

Understand that (f∘g)(x)=f(g(x)) (f \circ g)(x) = f(g(x)) (f∘g)(x)=f(g(x)).

STEP 4

Evaluate Option A:- f(x)=x2−1 f(x) = x^2 - 1 f(x)=x2−1 - g(x)=2x−1 g(x) = 2x - 1 g(x)=2x−1Substitute g(x) g(x) g(x) into f(x) f(x) f(x):f(g(x))=(2x−1)2−1 f(g(x)) = (2x - 1)^2 - 1 f(g(x))=(2x−1)2−1

STEP 5

STEP 6

Evaluate Option B:- f(x)=1−x2 f(x) = 1 - x^2 f(x)=1−x2 - g(x)=2x+1 g(x) = 2x + 1 g(x)=2x+1Substitute g(x) g(x) g(x) into f(x) f(x) f(x):f(g(x))=1−(2x+1)2 f(g(x)) = 1 - (2x + 1)^2 f(g(x))=1−(2x+1)2

STEP 7

STEP 8

Since Option B matches the given expression, we have found the correct functions:f(x)=1−x2 f(x) = 1 - x^2 f(x)=1−x2 g(x)=2x+1 g(x) = 2x + 1 g(x)=2x+1 The correct pair of functions is:B:f(x)=1−x2,g(x)=2x+1 \boxed{B: f(x) = 1 - x^2, g(x) = 2x + 1} B:f(x)=1−x2,g(x)=2x+1​

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 that $(f \circ g)(x) = -4x(x+1)$ .
2. We need to determine the functions $f(x)$ and $g(x)$ from the given options.

1. Understand the composition of functions.
2. Evaluate each option by substituting $g(x)$ into $f(x)$ .
3. Compare the result with the given expression $-4x(x+1)$ .
4. Identify the correct pair of functions.

Understand that $(f \circ g)(x) = f(g(x))$ .

Evaluate Option A:
- $f(x) = x^2 - 1$ - $g(x) = 2x - 1$
Substitute $g(x)$ into $f(x)$ :
$f(g(x)) = (2x - 1)^2 - 1$

Evaluate Option B:
- $f(x) = 1 - x^2$ - $g(x) = 2x + 1$
Substitute $g(x)$ into $f(x)$ :
$f(g(x)) = 1 - (2x + 1)^2$

Since Option B matches the given expression, we have found the correct functions:
$f(x) = 1 - x^2$ $g(x) = 2x + 1$
The correct pair of functions is:
$\boxed{B: f(x) = 1 - x^2, g(x) = 2x + 1}$