Math  /  Algebra

QuestionFor each pair of functions ff and gg below, find f(g(x))f(g(x)) and g(f(x))g(f(x)). Then, determine whether ff and gg are inverses of each other.
Simplify your answers as much as possible. (Assume that your expressions are defined for all xx in the domain of the composition. You do not have to indicate the domain.) (a) f(x)=2x,x0f(x)=\frac{2}{x}, x \neq 0 (b) f(x)=3x7f(x)=3 x-7 g(x)=2x,x0f(g(x))=xg(f(x))=x\begin{array}{l} g(x)=\frac{2}{x}, x \neq 0 \\ f(g(x))=x \\ g(f(x))=x \end{array} ff and gg are inverses of each other ff and gg are inverses of each other ff and gg are not inverses of each other ff and gg are not inverses of each other

Studdy Solution

STEP 1

What is this asking? We're checking if two function pairs are inverses of each other by plugging one into the other and simplifying. Watch out! Remember, a function and its inverse undo each other, so if they *are* inverses, plugging one into the other should simplify to just xx.

STEP 2

1. Tackle the first function pair.
2. Check out the second function pair.

STEP 3

Alright, let's **start** with f(x)=2xf(x) = \frac{2}{x} and g(x)=2xg(x) = \frac{2}{x}.
To find f(g(x))f(g(x)), we're gonna plug g(x)g(x) right into f(x)f(x)!
So, wherever we see an xx in f(x)f(x), we'll replace it with 2x\frac{2}{x}.
This gives us f(g(x))=22xf(g(x)) = \frac{2}{\frac{2}{x}}.

STEP 4

To simplify this funky fraction, remember dividing by a fraction is the same as multiplying by its reciprocal!
So, 22x\frac{2}{\frac{2}{x}} becomes 2x22 \cdot \frac{x}{2}.
The **twos** divide to one, leaving us with just xx!

STEP 5

Now, let's find g(f(x))g(f(x)).
We'll plug f(x)f(x) into g(x)g(x).
This looks just like the last one, since f(x)f(x) and g(x)g(x) are the same!
So, g(f(x))=22xg(f(x)) = \frac{2}{\frac{2}{x}}.

STEP 6

Just like before, we simplify the fraction: 22x=2x2=x\frac{2}{\frac{2}{x}} = 2 \cdot \frac{x}{2} = x.

STEP 7

Since f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x, these two *are* inverses of each other!

STEP 8

Now, we have f(x)=3x7f(x) = 3x - 7 and g(x)=2xg(x) = \frac{2}{x}.
Let's find f(g(x))f(g(x)) by plugging g(x)g(x) into f(x)f(x): f(g(x))=3(2x)7f(g(x)) = 3 \cdot \left(\frac{2}{x}\right) - 7.

STEP 9

This simplifies to f(g(x))=6x7f(g(x)) = \frac{6}{x} - 7.

STEP 10

Next, let's find g(f(x))g(f(x)) by plugging f(x)f(x) into g(x)g(x): g(f(x))=23x7g(f(x)) = \frac{2}{3x - 7}.

STEP 11

Since f(g(x))f(g(x)) and g(f(x))g(f(x)) don't simplify to xx, these two are *not* inverses of each other!

STEP 12

For the first pair, f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x, so they *are* inverses.
For the second pair, f(g(x))=6x7f(g(x)) = \frac{6}{x} - 7 and g(f(x))=23x7g(f(x)) = \frac{2}{3x - 7}, so they are *not* inverses.

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