QuestionWhich of the following have an inverse that is a function as well?
Studdy Solution
STEP 1
What is this asking? Which of these equations describe functions that have inverse functions? Watch out! A function only has an inverse function if it passes the horizontal line test!
STEP 2
1. Cubic Function
2. Quadratic Function
3. Linear Function
4. Constant Function
5. Rational Function
6. Exponential Function
STEP 3
Let's look at .
Cubic functions **generally** pass the horizontal line test, meaning any horizontal line only crosses the graph **once**.
This one does, so it **has** an inverse function!
STEP 4
Now for . **Quadratic functions (parabolas)** fail the horizontal line test.
Imagine a horizontal line intersecting a parabola—it crosses at two points!
So, **no inverse function** here.
STEP 5
Consider .
This is a **linear function**, and as long as the line isn't horizontal, it'll pass the horizontal line test!
This one has a slope of , so it's not horizontal, and thus **has an inverse function**.
STEP 6
is a **constant function**, which is just a horizontal line.
It definitely **fails** the horizontal line test, so **no inverse function**.
STEP 7
Time for .
This **rational function** has a horizontal asymptote at .
If we draw a horizontal line *anywhere else*, it'll only intersect the graph once!
So, it **passes** the horizontal line test, and therefore **has an inverse function**.
STEP 8
Finally, . **Exponential functions** also pass the horizontal line test!
They go up and up (or down and down if the base is between 0 and 1), but any horizontal line only crosses them **once**.
So, this one **has an inverse function**.
STEP 9
The functions , , , and all have inverse functions.
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