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) $f(x)=\frac{1}{2 x}, x \neq 0$ (b) $f(x)=x+6$ $\begin{array}{l} g(x)=\frac{1}{2 x}, x \neq 0 \\ f(g(x))=\square \\ g(f(x))=\square \end{array}$ $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other $\begin{array}{l} g(x)=x+6 \\ f(g(x))=\square \\ g(f(x))=\square \end{array}$ $f$ and $g$ are 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 the compositions $f(g(x))$ and $g(f(x))$ for each pair.
3. We need to determine if $f$ and $g$ are inverses of each other.

STEP 2

1. Evaluate $f(g(x))$ and $g(f(x))$ for the first pair of functions.
2. Determine if the first pair are inverses.
3. Evaluate $f(g(x))$ and $g(f(x))$ for the second pair of functions.
4. Determine if the second pair are inverses.

STEP 3

For the first pair, $f(x) = \frac{1}{2x}$ and $g(x) = \frac{1}{2x}$ .
Calculate $f(g(x))$ : $f(g(x)) = f\left(\frac{1}{2x}\right) = \frac{1}{2 \left(\frac{1}{2x}\right)} = x$
Calculate $g(f(x))$ : $g(f(x)) = g\left(\frac{1}{2x}\right) = \frac{1}{2 \left(\frac{1}{2x}\right)} = x$

STEP 4

Since $f(g(x)) = x$ and $g(f(x)) = x$ , $f$ and $g$ are inverses of each other.

STEP 5

For the second pair, $f(x) = x + 6$ and $g(x) = x + 6$ .
Calculate $f(g(x))$ : $f(g(x)) = f(x + 6) = (x + 6) + 6 = x + 12$
Calculate $g(f(x))$ : $g(f(x)) = g(x + 6) = (x + 6) + 6 = x + 12$

STEP 6

Since $f(g(x)) \neq x$ and $g(f(x)) \neq x$ , $f$ and $g$ are not inverses of each other.
The results are: (a) $f$ and $g$ are inverses of each other. (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 the compositions $f(g(x))$ and $g(f(x))$ for each pair.
3. We need to determine if $f$ and $g$ are inverses of each other.

STEP 2

1. Evaluate $f(g(x))$ and $g(f(x))$ for the first pair of functions.
2. Determine if the first pair are inverses.
3. Evaluate $f(g(x))$ and $g(f(x))$ for the second pair of functions.
4. Determine if the second pair are inverses.

STEP 3

For the first pair, $f(x) = \frac{1}{2x}$ and $g(x) = \frac{1}{2x}$ .
Calculate $f(g(x))$ : $f(g(x)) = f\left(\frac{1}{2x}\right) = \frac{1}{2 \left(\frac{1}{2x}\right)} = x$
Calculate $g(f(x))$ : $g(f(x)) = g\left(\frac{1}{2x}\right) = \frac{1}{2 \left(\frac{1}{2x}\right)} = x$

STEP 4

Since $f(g(x)) = x$ and $g(f(x)) = x$ , $f$ and $g$ are inverses of each other.

STEP 5

For the second pair, $f(x) = x + 6$ and $g(x) = x + 6$ .
Calculate $f(g(x))$ : $f(g(x)) = f(x + 6) = (x + 6) + 6 = x + 12$
Calculate $g(f(x))$ : $g(f(x)) = g(x + 6) = (x + 6) + 6 = x + 12$

STEP 6

Since $f(g(x)) \neq x$ and $g(f(x)) \neq x$ , $f$ and $g$ are not inverses of each other.
The results are: (a) $f$ and $g$ are inverses of each other. (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 the compositions 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.

STEP 2

1. Evaluate f(g(x)) f(g(x)) f(g(x)) and g(f(x)) g(f(x)) g(f(x)) for the first pair of functions.2. Determine if the first pair are inverses.3. Evaluate f(g(x)) f(g(x)) f(g(x)) and g(f(x)) g(f(x)) g(f(x)) for the second pair of functions.4. Determine if the second pair are inverses.

STEP 3

STEP 4

Since 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 f f and g g g are inverses of each other.

STEP 5

STEP 6

Since f(g(x))≠x f(g(x)) \neq x f(g(x))=x and g(f(x))≠x g(f(x)) \neq x g(f(x))=x, f f f and g g g are not inverses of each other. The results are: (a) f f f and g g g are inverses of each other. (b) f f f and g g g are not inverses of each other.

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1. We are given two pairs of functions $f$ and $g$ .
2. We need to find the compositions $f(g(x))$ and $g(f(x))$ for each pair.
3. We need to determine if $f$ and $g$ are inverses of each other.

1. Evaluate $f(g(x))$ and $g(f(x))$ for the first pair of functions.
2. Determine if the first pair are inverses.
3. Evaluate $f(g(x))$ and $g(f(x))$ for the second pair of functions.
4. Determine if the second pair are inverses.

Since $f(g(x)) = x$ and $g(f(x)) = x$ , $f$ and $g$ are inverses of each other.

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