Math / Algebra

QuestionFor $f(x)=\sqrt{x}$ and $g(x)=6 x+1$ , find the following composite functions and state the domain of each. (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ (d) $g \circ g$

Studdy Solution

STEP 1

1. We are given two functions: $f(x) = \sqrt{x}$ and $g(x) = 6x + 1$ .
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. Find the composite function $f \circ g$ and its domain.
2. Find the composite function $g \circ f$ and its domain.
3. Find the composite function $f \circ f$ and its domain.
4. Find the composite function $g \circ g$ and its domain.

STEP 3

Find the composite function $f \circ g$ :
$(f \circ g)(x) = f(g(x)) = f(6x + 1) = \sqrt{6x + 1}$

STEP 4

Determine the domain of $f \circ g$ :
The expression inside the square root, $6x + 1$ , must be non-negative:
$6x + 1 \geq 0$ $6x \geq -1$ $x \geq -\frac{1}{6}$
Thus, the domain of $f \circ g$ is $x \geq -\frac{1}{6}$ .

STEP 5

Find the composite function $g \circ f$ :
$(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 6\sqrt{x} + 1$

STEP 6

Determine the domain of $g \circ f$ :
The expression inside the square root, $x$ , must be non-negative:
$x \geq 0$
Thus, the domain of $g \circ f$ is $x \geq 0$ .

STEP 7

Find the composite function $f \circ f$ :
$(f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}$

STEP 8

Determine the domain of $f \circ f$ :
The expression inside the square root, $x$ , must be non-negative:
$x \geq 0$
Thus, the domain of $f \circ f$ is $x \geq 0$ .

STEP 9

Find the composite function $g \circ g$ :
$(g \circ g)(x) = g(g(x)) = g(6x + 1) = 6(6x + 1) + 1 = 36x + 6 + 1 = 36x + 7$

STEP 10

Determine the domain of $g \circ g$ :
Since $g \circ g$ is a linear function, its domain is all real numbers.
The composite functions and their domains are: (a) $f \circ g(x) = \sqrt{6x + 1}$ , domain: $x \geq -\frac{1}{6}$ (b) $g \circ f(x) = 6\sqrt{x} + 1$ , domain: $x \geq 0$ (c) $f \circ f(x) = x^{1/4}$ , domain: $x \geq 0$ (d) $g \circ g(x) = 36x + 7$ , domain: all real numbers

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QuestionFor $f(x)=\sqrt{x}$ and $g(x)=6 x+1$ , find the following composite functions and state the domain of each. (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ (d) $g \circ g$

Studdy Solution

STEP 1

1. We are given two functions: $f(x) = \sqrt{x}$ and $g(x) = 6x + 1$ .
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. Find the composite function $f \circ g$ and its domain.
2. Find the composite function $g \circ f$ and its domain.
3. Find the composite function $f \circ f$ and its domain.
4. Find the composite function $g \circ g$ and its domain.

STEP 3

Find the composite function $f \circ g$ :
$(f \circ g)(x) = f(g(x)) = f(6x + 1) = \sqrt{6x + 1}$

STEP 4

Determine the domain of $f \circ g$ :
The expression inside the square root, $6x + 1$ , must be non-negative:
$6x + 1 \geq 0$ $6x \geq -1$ $x \geq -\frac{1}{6}$
Thus, the domain of $f \circ g$ is $x \geq -\frac{1}{6}$ .

STEP 5

Find the composite function $g \circ f$ :
$(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 6\sqrt{x} + 1$

STEP 6

Determine the domain of $g \circ f$ :
The expression inside the square root, $x$ , must be non-negative:
$x \geq 0$
Thus, the domain of $g \circ f$ is $x \geq 0$ .

STEP 7

Find the composite function $f \circ f$ :
$(f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}$

STEP 8

Determine the domain of $f \circ f$ :
The expression inside the square root, $x$ , must be non-negative:
$x \geq 0$
Thus, the domain of $f \circ f$ is $x \geq 0$ .

STEP 9

Find the composite function $g \circ g$ :
$(g \circ g)(x) = g(g(x)) = g(6x + 1) = 6(6x + 1) + 1 = 36x + 6 + 1 = 36x + 7$

STEP 10

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QuestionFor f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=6x+1g(x)=6 x+1g(x)=6x+1, find the following composite functions and state the domain of each. (a) f∘gf \circ gf∘g (b) g∘fg \circ fg∘f (c) f∘ff \circ ff∘f (d) g∘gg \circ gg∘g

Studdy Solution

STEP 1

STEP 2

1. Find the composite function f∘g f \circ g f∘g and its domain.2. Find the composite function g∘f g \circ f g∘f and its domain.3. Find the composite function f∘f f \circ f f∘f and its domain.4. Find the composite function g∘g g \circ g g∘g and its domain.

STEP 3

Find the composite function f∘g f \circ g f∘g:(f∘g)(x)=f(g(x))=f(6x+1)=6x+1 (f \circ g)(x) = f(g(x)) = f(6x + 1) = \sqrt{6x + 1} (f∘g)(x)=f(g(x))=f(6x+1)=6x+1​

STEP 4

STEP 5

Find the composite function g∘f g \circ f g∘f:(g∘f)(x)=g(f(x))=g(x)=6x+1 (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 6\sqrt{x} + 1 (g∘f)(x)=g(f(x))=g(x​)=6x​+1

STEP 6

Determine the domain of g∘f g \circ f g∘f:The expression inside the square root, x x x, must be non-negative:x≥0 x \geq 0 x≥0Thus, the domain of g∘f g \circ f g∘f is x≥0 x \geq 0 x≥0.

STEP 7

Find the composite function 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 inside the square root, x x x, must be non-negative:x≥0 x \geq 0 x≥0Thus, the domain of f∘f f \circ f f∘f is x≥0 x \geq 0 x≥0.

STEP 9

Find the composite function g∘g g \circ g g∘g:(g∘g)(x)=g(g(x))=g(6x+1)=6(6x+1)+1=36x+6+1=36x+7 (g \circ g)(x) = g(g(x)) = g(6x + 1) = 6(6x + 1) + 1 = 36x + 6 + 1 = 36x + 7 (g∘g)(x)=g(g(x))=g(6x+1)=6(6x+1)+1=36x+6+1=36x+7

STEP 10

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QuestionFor $f(x)=\sqrt{x}$ and $g(x)=6 x+1$ , find the following composite functions and state the domain of each. (a) $f \circ g$ (b) $g \circ f$ (c) $f \circ f$ (d) $g \circ g$

1. Find the composite function $f \circ g$ and its domain.
2. Find the composite function $g \circ f$ and its domain.
3. Find the composite function $f \circ f$ and its domain.
4. Find the composite function $g \circ g$ and its domain.

Find the composite function $f \circ g$ :
$(f \circ g)(x) = f(g(x)) = f(6x + 1) = \sqrt{6x + 1}$

Find the composite function $g \circ f$ :
$(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 6\sqrt{x} + 1$

Determine the domain of $g \circ f$ :
The expression inside the square root, $x$ , must be non-negative:
$x \geq 0$
Thus, the domain of $g \circ f$ is $x \geq 0$ .

Find the composite function $f \circ f$ :
$(f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4}$

Determine the domain of $f \circ f$ :
The expression inside the square root, $x$ , must be non-negative:
$x \geq 0$
Thus, the domain of $f \circ f$ is $x \geq 0$ .

Find the composite function $g \circ g$ :
$(g \circ g)(x) = g(g(x)) = g(6x + 1) = 6(6x + 1) + 1 = 36x + 6 + 1 = 36x + 7$