Math / Algebra

QuestionFor $f(x)=\sqrt{x}$ and $g(x)=4 x+1$ , find the following composite functions and state the domain of each. (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ (d) $\mathrm{g} \circ \mathrm{g}$ (a) $(f \circ g)(x)=$ $\square$ (Simplify your answer.)
Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $f \circ \mathrm{~g}$ is $\{\mathrm{x} \mid \square\}$ . $\square$ (Type an inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The domain of $f \circ g$ is all real numbers. (b) $(g \circ f)(x)=$ $\square$ (Simplify your answer.)
Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $g \circ f$ is $\{x$ $\square$ \}. (Type an inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The domain of $g \circ f$ is all real numbers. (c) $(f \circ f)(x)=$ $\square$ (Simplify your answer.)
Select the correct choice below and fill in any answer boxes within your choice. A. The domain of $f$ o $f$ is $\{x$ $\square$ \}. (Type an inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) B. The domain of $f$ o $f$ is all real numbers.

Studdy Solution

STEP 1

1. We are given two functions: $f(x) = \sqrt{x}$ and $g(x) = 4x + 1$ .
2. We need to find the composite functions $f \circ g$ , $g \circ f$ , $f \circ f$ , and $g \circ g$ .
3. We need to determine the domain of each composite function.

STEP 2

1. Calculate $f \circ g$ and determine its domain.
2. Calculate $g \circ f$ and determine its domain.
3. Calculate $f \circ f$ and determine its domain.
4. Calculate $g \circ g$ and determine its domain.

STEP 3

Calculate $f \circ g$ :
$(f \circ g)(x) = f(g(x)) = f(4x + 1) = \sqrt{4x + 1}$

STEP 4

Determine the domain of $f \circ g$ :
The expression $\sqrt{4x + 1}$ is defined when $4x + 1 \geq 0$ .
$4x + 1 \geq 0$ $4x \geq -1$ $x \geq -\frac{1}{4}$
So, the domain of $f \circ g$ is:
$\{ x \mid x \geq -\frac{1}{4} \}$

STEP 5

Calculate $g \circ f$ :
$(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 4\sqrt{x} + 1$

STEP 6

Determine the domain of $g \circ f$ :
The expression $4\sqrt{x} + 1$ is defined when $\sqrt{x}$ is defined, which is when $x \geq 0$ .
So, the domain of $g \circ f$ is:
$\{ x \mid x \geq 0 \}$

STEP 7

Calculate $f \circ f$ :
$(f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}$

STEP 8

Determine the domain of $f \circ f$ :
The expression $x^{1/4}$ is defined when $x \geq 0$ .
So, the domain of $f \circ f$ is:
$\{ x \mid x \geq 0 \}$

STEP 9

Calculate $g \circ g$ :
$(g \circ g)(x) = g(g(x)) = g(4x + 1) = 4(4x + 1) + 1 = 16x + 4 + 1 = 16x + 5$

STEP 10

Determine the domain of $g \circ g$ :
The expression $16x + 5$ is a linear function, which is defined for all real numbers.
So, the domain of $g \circ g$ is all real numbers.
The solutions for the composite functions and their domains are:
(a) $(f \circ g)(x) = \sqrt{4x + 1}$ , Domain: $\{ x \mid x \geq -\frac{1}{4} \}$
(b) $(g \circ f)(x) = 4\sqrt{x} + 1$ , Domain: $\{ x \mid x \geq 0 \}$
(c) $(f \circ f)(x) = x^{1/4}$ , Domain: $\{ x \mid x \geq 0 \}$
(d) $(g \circ g)(x) = 16x + 5$ , Domain: All real numbers

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Studdy Solution

STEP 1

1. We are given two functions: $f(x) = \sqrt{x}$ and $g(x) = 4x + 1$ .
2. We need to find the composite functions $f \circ g$ , $g \circ f$ , $f \circ f$ , and $g \circ g$ .
3. We need to determine the domain of each composite function.

STEP 2

1. Calculate $f \circ g$ and determine its domain.
2. Calculate $g \circ f$ and determine its domain.
3. Calculate $f \circ f$ and determine its domain.
4. Calculate $g \circ g$ and determine its domain.

STEP 3

Calculate $f \circ g$ :
$(f \circ g)(x) = f(g(x)) = f(4x + 1) = \sqrt{4x + 1}$

STEP 4

Determine the domain of $f \circ g$ :
The expression $\sqrt{4x + 1}$ is defined when $4x + 1 \geq 0$ .
$4x + 1 \geq 0$ $4x \geq -1$ $x \geq -\frac{1}{4}$
So, the domain of $f \circ g$ is:
$\{ x \mid x \geq -\frac{1}{4} \}$

STEP 5

Calculate $g \circ f$ :
$(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 4\sqrt{x} + 1$

STEP 6

Determine the domain of $g \circ f$ :
The expression $4\sqrt{x} + 1$ is defined when $\sqrt{x}$ is defined, which is when $x \geq 0$ .
So, the domain of $g \circ f$ is:
$\{ x \mid x \geq 0 \}$

STEP 7

Calculate $f \circ f$ :
$(f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}$

STEP 8

Determine the domain of $f \circ f$ :
The expression $x^{1/4}$ is defined when $x \geq 0$ .
So, the domain of $f \circ f$ is:
$\{ x \mid x \geq 0 \}$

STEP 9

Calculate $g \circ g$ :
$(g \circ g)(x) = g(g(x)) = g(4x + 1) = 4(4x + 1) + 1 = 16x + 4 + 1 = 16x + 5$

STEP 10

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Studdy Solution

STEP 1

STEP 2

1. Calculate f∘g f \circ g f∘g and determine its domain.2. Calculate g∘f g \circ f g∘f and determine its domain.3. Calculate f∘f f \circ f f∘f and determine its domain.4. Calculate g∘g g \circ g g∘g and determine its domain.

STEP 3

Calculate f∘g f \circ g f∘g:(f∘g)(x)=f(g(x))=f(4x+1)=4x+1 (f \circ g)(x) = f(g(x)) = f(4x + 1) = \sqrt{4x + 1} (f∘g)(x)=f(g(x))=f(4x+1)=4x+1​

STEP 4

STEP 5

Calculate g∘f g \circ f g∘f:(g∘f)(x)=g(f(x))=g(x)=4x+1 (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 4\sqrt{x} + 1 (g∘f)(x)=g(f(x))=g(x​)=4x​+1

STEP 6

Determine the domain of g∘f g \circ f g∘f:The expression 4x+1 4\sqrt{x} + 1 4x​+1 is defined when x \sqrt{x} x​ is defined, which is when x≥0 x \geq 0 x≥0.So, the domain of g∘f g \circ f g∘f is:{x∣x≥0} \{ x \mid x \geq 0 \} {x∣x≥0}

STEP 7

Calculate f∘f f \circ f f∘f:(f∘f)(x)=f(f(x))=f(x)=x=x1/4 (f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4} (f∘f)(x)=f(f(x))=f(x​)=x​​=x1/4

STEP 8

Determine the domain of f∘f f \circ f f∘f:The expression x1/4 x^{1/4} x1/4 is defined when x≥0 x \geq 0 x≥0.So, the domain of f∘f f \circ f f∘f is:{x∣x≥0} \{ x \mid x \geq 0 \} {x∣x≥0}

STEP 9

Calculate g∘g g \circ g g∘g:(g∘g)(x)=g(g(x))=g(4x+1)=4(4x+1)+1=16x+4+1=16x+5 (g \circ g)(x) = g(g(x)) = g(4x + 1) = 4(4x + 1) + 1 = 16x + 4 + 1 = 16x + 5 (g∘g)(x)=g(g(x))=g(4x+1)=4(4x+1)+1=16x+4+1=16x+5

STEP 10

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1. Calculate $f \circ g$ and determine its domain.
2. Calculate $g \circ f$ and determine its domain.
3. Calculate $f \circ f$ and determine its domain.
4. Calculate $g \circ g$ and determine its domain.

Calculate $f \circ g$ :
$(f \circ g)(x) = f(g(x)) = f(4x + 1) = \sqrt{4x + 1}$

Calculate $g \circ f$ :
$(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 4\sqrt{x} + 1$

Determine the domain of $g \circ f$ :
The expression $4\sqrt{x} + 1$ is defined when $\sqrt{x}$ is defined, which is when $x \geq 0$ .
So, the domain of $g \circ f$ is:
$\{ x \mid x \geq 0 \}$

Calculate $f \circ f$ :
$(f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}$

Determine the domain of $f \circ f$ :
The expression $x^{1/4}$ is defined when $x \geq 0$ .
So, the domain of $f \circ f$ is:
$\{ x \mid x \geq 0 \}$

Calculate $g \circ g$ :
$(g \circ g)(x) = g(g(x)) = g(4x + 1) = 4(4x + 1) + 1 = 16x + 4 + 1 = 16x + 5$