Math / Algebra

QuestionGiven the function $h(x) = (x + 2)^2$ , find functions $f(x)$ and $g(x)$ such that $h(x) = f(g(x))$ .

Studdy Solution

STEP 1

1. The function $h(x) = (x + 2)^2$ is given.
2. We need to find two functions $f(x)$ and $g(x)$ such that their composition $f(g(x))$ results in $h(x)$ .
3. The composition of functions means that $f(g(x)) = h(x)$ .

STEP 2

1. Identify an inner function $g(x)$ that can be used inside another function.
2. Determine the outer function $f(x)$ that, when applied to $g(x)$ , results in $h(x)$ .
3. Verify that the composition $f(g(x))$ correctly reproduces $h(x)$ .

STEP 3

Choose an inner function $g(x)$ that simplifies part of the expression $h(x) = (x + 2)^2$ .
Let $g(x) = x + 2$ .

STEP 4

Determine the outer function $f(x)$ that, when applied to $g(x)$ , results in $h(x)$ .
Since $h(x) = (g(x))^2$ and $g(x) = x + 2$ , we let $f(x) = x^2$ .

STEP 5

Verify that the composition $f(g(x))$ correctly reproduces $h(x)$ .
Calculate $f(g(x))$ : $f(g(x)) = f(x + 2) = (x + 2)^2$
Since this matches $h(x)$ , the functions $f(x) = x^2$ and $g(x) = x + 2$ are correct.
Solution: The functions are $f(x) = x^2$ and $g(x) = x + 2$ .

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QuestionGiven the function $h(x) = (x + 2)^2$ , find functions $f(x)$ and $g(x)$ such that $h(x) = f(g(x))$ .

Studdy Solution

STEP 1

1. The function $h(x) = (x + 2)^2$ is given.
2. We need to find two functions $f(x)$ and $g(x)$ such that their composition $f(g(x))$ results in $h(x)$ .
3. The composition of functions means that $f(g(x)) = h(x)$ .

STEP 2

1. Identify an inner function $g(x)$ that can be used inside another function.
2. Determine the outer function $f(x)$ that, when applied to $g(x)$ , results in $h(x)$ .
3. Verify that the composition $f(g(x))$ correctly reproduces $h(x)$ .

STEP 3

Choose an inner function $g(x)$ that simplifies part of the expression $h(x) = (x + 2)^2$ .
Let $g(x) = x + 2$ .

STEP 4

Determine the outer function $f(x)$ that, when applied to $g(x)$ , results in $h(x)$ .
Since $h(x) = (g(x))^2$ and $g(x) = x + 2$ , we let $f(x) = x^2$ .

STEP 5

Verify that the composition $f(g(x))$ correctly reproduces $h(x)$ .
Calculate $f(g(x))$ : $f(g(x)) = f(x + 2) = (x + 2)^2$
Since this matches $h(x)$ , the functions $f(x) = x^2$ and $g(x) = x + 2$ are correct.
Solution: The functions are $f(x) = x^2$ and $g(x) = x + 2$ .

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QuestionGiven the function h(x)=(x+2)2 h(x) = (x + 2)^2 h(x)=(x+2)2, find functions f(x) f(x) f(x) and g(x) g(x) g(x) such that h(x)=f(g(x)) h(x) = f(g(x)) h(x)=f(g(x)).

Studdy Solution

STEP 1

1. The function h(x)=(x+2)2 h(x) = (x + 2)^2 h(x)=(x+2)2 is given.2. We need to find two functions f(x) f(x) f(x) and g(x) g(x) g(x) such that their composition f(g(x)) f(g(x)) f(g(x)) results in h(x) h(x) h(x).3. The composition of functions means that f(g(x))=h(x) f(g(x)) = h(x) f(g(x))=h(x).

STEP 2

1. Identify an inner function g(x) g(x) g(x) that can be used inside another function.2. Determine the outer function f(x) f(x) f(x) that, when applied to g(x) g(x) g(x), results in h(x) h(x) h(x).3. Verify that the composition f(g(x)) f(g(x)) f(g(x)) correctly reproduces h(x) h(x) h(x).

STEP 3

Choose an inner function g(x) g(x) g(x) that simplifies part of the expression h(x)=(x+2)2 h(x) = (x + 2)^2 h(x)=(x+2)2.Let g(x)=x+2 g(x) = x + 2 g(x)=x+2.

STEP 4

Determine the outer function f(x) f(x) f(x) that, when applied to g(x) g(x) g(x), results in h(x) h(x) h(x).Since h(x)=(g(x))2 h(x) = (g(x))^2 h(x)=(g(x))2 and g(x)=x+2 g(x) = x + 2 g(x)=x+2, we let f(x)=x2 f(x) = x^2 f(x)=x2.

STEP 5

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QuestionGiven the function $h(x) = (x + 2)^2$ , find functions $f(x)$ and $g(x)$ such that $h(x) = f(g(x))$ .

1. The function $h(x) = (x + 2)^2$ is given.
2. We need to find two functions $f(x)$ and $g(x)$ such that their composition $f(g(x))$ results in $h(x)$ .
3. The composition of functions means that $f(g(x)) = h(x)$ .

1. Identify an inner function $g(x)$ that can be used inside another function.
2. Determine the outer function $f(x)$ that, when applied to $g(x)$ , results in $h(x)$ .
3. Verify that the composition $f(g(x))$ correctly reproduces $h(x)$ .

Choose an inner function $g(x)$ that simplifies part of the expression $h(x) = (x + 2)^2$ .
Let $g(x) = x + 2$ .

Determine the outer function $f(x)$ that, when applied to $g(x)$ , results in $h(x)$ .
Since $h(x) = (g(x))^2$ and $g(x) = x + 2$ , we let $f(x) = x^2$ .