Math  /  Algebra

QuestionWhich of the following have an inverse that is a function as well? f(x)=2x3+4f(x) = 2x^3 + 4 f(x)=x25x+4f(x) = -x^2 - 5x + 4 f(x)=2xf(x) = 2x f(x)=4f(x) = 4 f(x)=2x+25f(x) = \frac{-2}{x+2} - 5 f(x)=2xf(x) = 2^x

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 f(x)=2x3+4f(x) = 2x^3 + 4.
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 f(x)=x25x+4f(x) = -x^2 - 5x + 4. **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 f(x)=2xf(x) = 2x.
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 2\mathbf{2}, so it's not horizontal, and thus **has an inverse function**.

STEP 6

f(x)=4f(x) = 4 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 f(x)=2x+25f(x) = \frac{-2}{x+2} - 5.
This **rational function** has a horizontal asymptote at y=5y = -5.
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, f(x)=2xf(x) = 2^x. **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 f(x)=2x3+4f(x) = 2x^3 + 4, f(x)=2xf(x) = 2x, f(x)=2x+25f(x) = \frac{-2}{x+2} - 5, and f(x)=2xf(x) = 2^x all have inverse functions.

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