Math / Algebra

QuestionFor each pair of functions $f$ and $g$ below, find $f(g(x))$ and $g(f(x))$ . Then, determine whether $f$ and $g$ are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all $x$ in the domain of the composition. You do not have to indicate the domain.) (a) $f(x)=-\frac{2}{x}, x \neq 0$ (b) $f(x)=3 x+5$ $\begin{array}{l} g(x)=-\frac{2}{x}, x \neq 0 \\ f(g(x))=\square \\ g(f(x))=\square \end{array}$ $f$ and $g$ are inverses of each other $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other $f$ and $g$ are not inverses of each other

Studdy Solution

STEP 1

1. We are given two pairs of functions $f$ and $g$ .
2. We need to find $f(g(x))$ and $g(f(x))$ for each pair.
3. We need to determine if $f$ and $g$ are inverses of each other.
4. The expressions are defined for all $x$ in the domain of the composition.

STEP 2

1. Evaluate $f(g(x))$ for each pair of functions.
2. Evaluate $g(f(x))$ for each pair of functions.
3. Determine if $f(x)$ and $g(x)$ are inverses by checking if $f(g(x)) = x$ and $g(f(x)) = x$ .

STEP 3

For pair (a), where $f(x) = -\frac{2}{x}$ and $g(x) = -\frac{2}{x}$ :
Evaluate $f(g(x))$ :
$f(g(x)) = f\left(-\frac{2}{x}\right) = -\frac{2}{\left(-\frac{2}{x}\right)}$
Simplify:
$f(g(x)) = -\frac{2 \cdot x}{-2} = x$

STEP 4

For pair (a), evaluate $g(f(x))$ :
$g(f(x)) = g\left(-\frac{2}{x}\right) = -\frac{2}{\left(-\frac{2}{x}\right)}$
Simplify:
$g(f(x)) = -\frac{2 \cdot x}{-2} = x$

STEP 5

For pair (a), since both $f(g(x)) = x$ and $g(f(x)) = x$ , $f(x)$ and $g(x)$ are inverses of each other.

STEP 6

For pair (b), where $f(x) = 3x + 5$ and $g(x) = -\frac{2}{x}$ :
Evaluate $f(g(x))$ :
$f(g(x)) = f\left(-\frac{2}{x}\right) = 3\left(-\frac{2}{x}\right) + 5$
Simplify:
$f(g(x)) = -\frac{6}{x} + 5$

STEP 7

For pair (b), evaluate $g(f(x))$ :
$g(f(x)) = g(3x + 5) = -\frac{2}{3x + 5}$

STEP 8

For pair (b), since $f(g(x)) \neq x$ and $g(f(x)) \neq x$ , $f(x)$ and $g(x)$ are not inverses of each other.
The results are: - For pair (a), $f$ and $g$ are inverses of each other. - For pair (b), $f$ and $g$ are not inverses of each other.

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Studdy Solution

STEP 1

1. We are given two pairs of functions $f$ and $g$ .
2. We need to find $f(g(x))$ and $g(f(x))$ for each pair.
3. We need to determine if $f$ and $g$ are inverses of each other.
4. The expressions are defined for all $x$ in the domain of the composition.

STEP 2

1. Evaluate $f(g(x))$ for each pair of functions.
2. Evaluate $g(f(x))$ for each pair of functions.
3. Determine if $f(x)$ and $g(x)$ are inverses by checking if $f(g(x)) = x$ and $g(f(x)) = x$ .

STEP 3

For pair (a), where $f(x) = -\frac{2}{x}$ and $g(x) = -\frac{2}{x}$ :
Evaluate $f(g(x))$ :
$f(g(x)) = f\left(-\frac{2}{x}\right) = -\frac{2}{\left(-\frac{2}{x}\right)}$
Simplify:
$f(g(x)) = -\frac{2 \cdot x}{-2} = x$

STEP 4

For pair (a), evaluate $g(f(x))$ :
$g(f(x)) = g\left(-\frac{2}{x}\right) = -\frac{2}{\left(-\frac{2}{x}\right)}$
Simplify:
$g(f(x)) = -\frac{2 \cdot x}{-2} = x$

STEP 5

For pair (a), since both $f(g(x)) = x$ and $g(f(x)) = x$ , $f(x)$ and $g(x)$ are inverses of each other.

STEP 6

For pair (b), where $f(x) = 3x + 5$ and $g(x) = -\frac{2}{x}$ :
Evaluate $f(g(x))$ :
$f(g(x)) = f\left(-\frac{2}{x}\right) = 3\left(-\frac{2}{x}\right) + 5$
Simplify:
$f(g(x)) = -\frac{6}{x} + 5$

STEP 7

For pair (b), evaluate $g(f(x))$ :
$g(f(x)) = g(3x + 5) = -\frac{2}{3x + 5}$

STEP 8

For pair (b), since $f(g(x)) \neq x$ and $g(f(x)) \neq x$ , $f(x)$ and $g(x)$ are not inverses of each other.
The results are: - For pair (a), $f$ and $g$ are inverses of each other. - For pair (b), $f$ and $g$ are not inverses of each other.

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Studdy Solution

STEP 1

1. We are given two pairs of functions f f f and g g g.2. We need to find f(g(x)) f(g(x)) f(g(x)) and g(f(x)) g(f(x)) g(f(x)) for each pair.3. We need to determine if f f f and g g g are inverses of each other.4. The expressions are defined for all x x x in the domain of the composition.

STEP 2

1. Evaluate f(g(x)) f(g(x)) f(g(x)) for each pair of functions.2. Evaluate g(f(x)) g(f(x)) g(f(x)) for each pair of functions.3. Determine if f(x) f(x) f(x) and g(x) g(x) g(x) are inverses by checking if 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.

STEP 3

STEP 4

For pair (a), evaluate g(f(x)) g(f(x)) g(f(x)):g(f(x))=g(−2x)=−2(−2x) g(f(x)) = g\left(-\frac{2}{x}\right) = -\frac{2}{\left(-\frac{2}{x}\right)} g(f(x))=g(−x2​)=−(−x2​)2​Simplify:g(f(x))=−2⋅x−2=x g(f(x)) = -\frac{2 \cdot x}{-2} = x g(f(x))=−−22⋅x​=x

STEP 5

For pair (a), 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, f(x) f(x) f(x) and g(x) g(x) g(x) are inverses of each other.

STEP 6

STEP 7

For pair (b), evaluate g(f(x)) g(f(x)) g(f(x)):g(f(x))=g(3x+5)=−23x+5 g(f(x)) = g(3x + 5) = -\frac{2}{3x + 5} g(f(x))=g(3x+5)=−3x+52​

STEP 8

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1. We are given two pairs of functions $f$ and $g$ .
2. We need to find $f(g(x))$ and $g(f(x))$ for each pair.
3. We need to determine if $f$ and $g$ are inverses of each other.
4. The expressions are defined for all $x$ in the domain of the composition.

1. Evaluate $f(g(x))$ for each pair of functions.
2. Evaluate $g(f(x))$ for each pair of functions.
3. Determine if $f(x)$ and $g(x)$ are inverses by checking if $f(g(x)) = x$ and $g(f(x)) = x$ .

For pair (a), evaluate $g(f(x))$ :
$g(f(x)) = g\left(-\frac{2}{x}\right) = -\frac{2}{\left(-\frac{2}{x}\right)}$
Simplify:
$g(f(x)) = -\frac{2 \cdot x}{-2} = x$

For pair (a), since both $f(g(x)) = x$ and $g(f(x)) = x$ , $f(x)$ and $g(x)$ are inverses of each other.

For pair (b), evaluate $g(f(x))$ :
$g(f(x)) = g(3x + 5) = -\frac{2}{3x + 5}$