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Math  /  Algebra

QuestionFor 30−3330-33, find the inverse function.
30. f(x)=x7f(x)=x^{7}
31. f(x)=132x5f(x)=\frac{1}{32} x^{5}
32. f(x)=−2x2+4f(x)=-2 x^{2}+4

Studdy Solution

STEP 1

What is this asking? We're on a mission to find the *inverse* of three different functions, which means figuring out how to "undo" what each function does! Watch out! Remember, not all functions have inverses!
We need to make sure these functions pass the horizontal line test.
Also, pay close attention to those exponents and coefficients – they can be tricky!

STEP 2

1. Inverse of f(x)=x7f(x) = x^7
2. Inverse of f(x)=132x5f(x) = \frac{1}{32}x^5
3. Inverse of f(x)=−2x2+4f(x) = -2x^2 + 4

STEP 3

**Define the function**: We're starting with f(x)=x7f(x) = x^7.
This takes an input xx and raises it to the **7th power**.

STEP 4

**Swap x and y**: To find the inverse, we **swap** xx and yy, getting x=y7x = y^7.
Think of it like switching the input and output!

STEP 5

**Solve for y**: Now, we need to **solve for** yy.
Since yy is raised to the 7th power, we'll take the **7th root** of both sides: x7=y77\sqrt[7]{x} = \sqrt[7]{y^7}.
This gives us y=x7y = \sqrt[7]{x}.

STEP 6

**Write the inverse**: So, the inverse function is f−1(x)=x7f^{-1}(x) = \sqrt[7]{x}!

STEP 7

**Define the function**: Our function is f(x)=132x5f(x) = \frac{1}{32}x^5.
It takes xx, raises it to the **5th power**, and then multiplies by 132\frac{1}{\textbf{32}}.

STEP 8

**Swap x and y**: Let's **swap** xx and yy: x=132y5x = \frac{1}{32}y^5.

STEP 9

**Solve for y**: To **isolate** yy, we first **multiply** both sides by **32** to get 32x=y532x = y^5.
Now, we take the **5th root** of both sides: 32x5=y55\sqrt[5]{32x} = \sqrt[5]{y^5}, which simplifies to y=32x5y = \sqrt[5]{32x}.
We can simplify this further by recognizing that 32=2532 = 2^5, so y=25⋅x5=2x5y = \sqrt[5]{2^5 \cdot x} = 2\sqrt[5]{x}.

STEP 10

**Write the inverse**: The inverse function is f−1(x)=2x5f^{-1}(x) = 2\sqrt[5]{x}!

STEP 11

**Define the function**: We have f(x)=−2x2+4f(x) = -2x^2 + 4.
This function squares the input xx, multiplies by −2-2, and then adds 44.

STEP 12

**Restrict the domain**: Uh oh!
This is a parabola, which doesn't pass the horizontal line test!
To make it invertible, we need to **restrict the domain**.
Let's choose x≥0x \ge 0.

STEP 13

**Swap x and y**: **Swapping** xx and yy gives us x=−2y2+4x = -2y^2 + 4.

STEP 14

**Solve for y**: Let's **isolate** yy.
First, subtract 44 from both sides: x−4=−2y2x - 4 = -2y^2.
Then, divide by −2-2: x−4−2=y2\frac{x - 4}{-2} = y^2, which simplifies to 4−x2=y2\frac{4 - x}{2} = y^2.
Taking the square root (remembering our domain restriction!) gives us y=4−x2y = \sqrt{\frac{4 - x}{2}}.

STEP 15

**Write the inverse**: The inverse function is f−1(x)=4−x2f^{-1}(x) = \sqrt{\frac{4 - x}{2}} for x≤4x \le 4.
Notice the domain restriction on x comes from the original range of f(x)f(x).

STEP 16

The inverse functions are: f−1(x)=x7f^{-1}(x) = \sqrt[7]{x} f−1(x)=2x5f^{-1}(x) = 2\sqrt[5]{x}f−1(x)=4−x2f^{-1}(x) = \sqrt{\frac{4 - x}{2}} for x≤4x \le 4

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