Math / Algebra

QuestionConsider the functions $f(x)=x^{3}-9$ and $g(x)=\sqrt[3]{x+9}$ (a) Find $f(g(x))$ . (b) Find $g(f(x))$ . (c) Determine whether the functions $f$ and $g$ are inverses of each other.

Studdy Solution

STEP 1

1. The function $f(x) = x^3 - 9$ is a cubic function.
2. The function $g(x) = \sqrt[3]{x+9}$ is a cube root function.
3. To determine if two functions are inverses, $f(g(x))$ and $g(f(x))$ should both equal $x$ .

STEP 2

1. Calculate $f(g(x))$ .
2. Calculate $g(f(x))$ .
3. Determine if $f$ and $g$ are inverses by checking if both compositions equal $x$ .

STEP 3

To find $f(g(x))$ , substitute $g(x) = \sqrt[3]{x+9}$ into $f(x) = x^3 - 9$ .
$f(g(x)) = f(\sqrt[3]{x+9}) = (\sqrt[3]{x+9})^3 - 9$

STEP 4

Simplify $f(g(x))$ :
$(\sqrt[3]{x+9})^3 = x + 9$
Thus,
$f(g(x)) = x + 9 - 9 = x$

STEP 5

To find $g(f(x))$ , substitute $f(x) = x^3 - 9$ into $g(x) = \sqrt[3]{x+9}$ .
$g(f(x)) = g(x^3 - 9) = \sqrt[3]{(x^3 - 9) + 9}$

STEP 6

Simplify $g(f(x))$ :
$\sqrt[3]{(x^3 - 9) + 9} = \sqrt[3]{x^3} = x$

STEP 7

Since both $f(g(x)) = x$ and $g(f(x)) = x$ , the functions $f$ and $g$ are inverses of each other.
The functions $f(x) = x^3 - 9$ and $g(x) = \sqrt[3]{x+9}$ are inverses of each other.

Was this helpful?

QuestionConsider the functions $f(x)=x^{3}-9$ and $g(x)=\sqrt[3]{x+9}$ (a) Find $f(g(x))$ . (b) Find $g(f(x))$ . (c) Determine whether the functions $f$ and $g$ are inverses of each other.

Studdy Solution

STEP 1

1. The function $f(x) = x^3 - 9$ is a cubic function.
2. The function $g(x) = \sqrt[3]{x+9}$ is a cube root function.
3. To determine if two functions are inverses, $f(g(x))$ and $g(f(x))$ should both equal $x$ .

STEP 2

1. Calculate $f(g(x))$ .
2. Calculate $g(f(x))$ .
3. Determine if $f$ and $g$ are inverses by checking if both compositions equal $x$ .

STEP 3

To find $f(g(x))$ , substitute $g(x) = \sqrt[3]{x+9}$ into $f(x) = x^3 - 9$ .
$f(g(x)) = f(\sqrt[3]{x+9}) = (\sqrt[3]{x+9})^3 - 9$

STEP 4

Simplify $f(g(x))$ :
$(\sqrt[3]{x+9})^3 = x + 9$
Thus,
$f(g(x)) = x + 9 - 9 = x$

STEP 5

To find $g(f(x))$ , substitute $f(x) = x^3 - 9$ into $g(x) = \sqrt[3]{x+9}$ .
$g(f(x)) = g(x^3 - 9) = \sqrt[3]{(x^3 - 9) + 9}$

STEP 6

Simplify $g(f(x))$ :
$\sqrt[3]{(x^3 - 9) + 9} = \sqrt[3]{x^3} = x$

STEP 7

Since both $f(g(x)) = x$ and $g(f(x)) = x$ , the functions $f$ and $g$ are inverses of each other.
The functions $f(x) = x^3 - 9$ and $g(x) = \sqrt[3]{x+9}$ are inverses of each other.

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.

QuestionConsider the functions f(x)=x3−9f(x)=x^{3}-9f(x)=x3−9 and g(x)=x+93g(x)=\sqrt[3]{x+9}g(x)=3x+9​ (a) Find f(g(x))f(g(x))f(g(x)). (b) Find g(f(x))g(f(x))g(f(x)). (c) Determine whether the functions fff and ggg are inverses of each other.

Studdy Solution

STEP 1

1. The function f(x)=x3−9 f(x) = x^3 - 9 f(x)=x3−9 is a cubic function.2. The function g(x)=x+93 g(x) = \sqrt[3]{x+9} g(x)=3x+9​ is a cube root function.3. To determine if two functions are inverses, f(g(x)) f(g(x)) f(g(x)) and g(f(x)) g(f(x)) g(f(x)) should both equal x x x.

STEP 2

1. Calculate f(g(x)) f(g(x)) f(g(x)).2. Calculate g(f(x)) g(f(x)) g(f(x)).3. Determine if f f f and g g g are inverses by checking if both compositions equal x x x.

STEP 3

To find f(g(x)) f(g(x)) f(g(x)), substitute g(x)=x+93 g(x) = \sqrt[3]{x+9} g(x)=3x+9​ into f(x)=x3−9 f(x) = x^3 - 9 f(x)=x3−9.f(g(x))=f(x+93)=(x+93)3−9 f(g(x)) = f(\sqrt[3]{x+9}) = (\sqrt[3]{x+9})^3 - 9 f(g(x))=f(3x+9​)=(3x+9​)3−9

STEP 4

Simplify f(g(x)) f(g(x)) f(g(x)):(x+93)3=x+9 (\sqrt[3]{x+9})^3 = x + 9 (3x+9​)3=x+9Thus,f(g(x))=x+9−9=x f(g(x)) = x + 9 - 9 = x f(g(x))=x+9−9=x

STEP 5

To find g(f(x)) g(f(x)) g(f(x)), substitute f(x)=x3−9 f(x) = x^3 - 9 f(x)=x3−9 into g(x)=x+93 g(x) = \sqrt[3]{x+9} g(x)=3x+9​.g(f(x))=g(x3−9)=(x3−9)+93 g(f(x)) = g(x^3 - 9) = \sqrt[3]{(x^3 - 9) + 9} g(f(x))=g(x3−9)=3(x3−9)+9​

STEP 6

Simplify g(f(x)) g(f(x)) g(f(x)):(x3−9)+93=x33=x \sqrt[3]{(x^3 - 9) + 9} = \sqrt[3]{x^3} = x 3(x3−9)+9​=3x3​=x

STEP 7

Since both f(g(x))=x f(g(x)) = x f(g(x))=x and g(f(x))=x g(f(x)) = x g(f(x))=x, the functions f f f and g g g are inverses of each other. The functions f(x)=x3−9 f(x) = x^3 - 9 f(x)=x3−9 and g(x)=x+93 g(x) = \sqrt[3]{x+9} g(x)=3x+9​ are inverses of each other.

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.

QuestionConsider the functions $f(x)=x^{3}-9$ and $g(x)=\sqrt[3]{x+9}$ (a) Find $f(g(x))$ . (b) Find $g(f(x))$ . (c) Determine whether the functions $f$ and $g$ are inverses of each other.

1. The function $f(x) = x^3 - 9$ is a cubic function.
2. The function $g(x) = \sqrt[3]{x+9}$ is a cube root function.
3. To determine if two functions are inverses, $f(g(x))$ and $g(f(x))$ should both equal $x$ .

1. Calculate $f(g(x))$ .
2. Calculate $g(f(x))$ .
3. Determine if $f$ and $g$ are inverses by checking if both compositions equal $x$ .

To find $f(g(x))$ , substitute $g(x) = \sqrt[3]{x+9}$ into $f(x) = x^3 - 9$ .
$f(g(x)) = f(\sqrt[3]{x+9}) = (\sqrt[3]{x+9})^3 - 9$

Simplify $f(g(x))$ :
$(\sqrt[3]{x+9})^3 = x + 9$
Thus,
$f(g(x)) = x + 9 - 9 = x$

To find $g(f(x))$ , substitute $f(x) = x^3 - 9$ into $g(x) = \sqrt[3]{x+9}$ .
$g(f(x)) = g(x^3 - 9) = \sqrt[3]{(x^3 - 9) + 9}$

Simplify $g(f(x))$ :
$\sqrt[3]{(x^3 - 9) + 9} = \sqrt[3]{x^3} = x$

Since both $f(g(x)) = x$ and $g(f(x)) = x$ , the functions $f$ and $g$ are inverses of each other.
The functions $f(x) = x^3 - 9$ and $g(x) = \sqrt[3]{x+9}$ are inverses of each other.