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Math  /  Algebra

QuestionThe function f(x)=x+24x−6f(x)=\frac{x+24}{x-6} is one-to-one. For the function, a. Find an equation for f−1(x)f^{-1}(x), the inverse function. b. Verify that your equation is correct by showing that f(f−1(x))=xf\left(f^{-1}(x)\right)=x and f−1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f−1(x)=f^{-1}(x)= □\square , for x≠\mathrm{x} \neq □\square B. f−1(x)=f^{-1}(x)= □\square , for x≤x \leq □\square C. f−1(x)=f^{-1}(x)= □\square , for x≥x \geq □\square D. f−1(x)=f^{-1}(x)= □\square , for all xx

Studdy Solution

STEP 1

1. We are given the function f(x)=x+24x−6 f(x) = \frac{x+24}{x-6} .
2. The function is one-to-one, which means it has an inverse.
3. We need to find the inverse function f−1(x) f^{-1}(x) .
4. We need to verify the inverse function by showing that f(f−1(x))=x f(f^{-1}(x)) = x and f−1(f(x))=x f^{-1}(f(x)) = x .

STEP 2

1. Find the inverse function f−1(x) f^{-1}(x) .
2. Verify that f(f−1(x))=x f(f^{-1}(x)) = x .
3. Verify that f−1(f(x))=x f^{-1}(f(x)) = x .
4. Choose the correct option for the inverse function.

STEP 3

To find the inverse function, start by setting y=f(x) y = f(x) :
y=x+24x−6 y = \frac{x+24}{x-6}

STEP 4

Solve for x x in terms of y y :
Multiply both sides by x−6 x - 6 :
y(x−6)=x+24 y(x - 6) = x + 24
Expand the left side:
yx−6y=x+24 yx - 6y = x + 24
Rearrange to isolate terms involving x x :
yx−x=6y+24 yx - x = 6y + 24
Factor out x x on the left side:
x(y−1)=6y+24 x(y - 1) = 6y + 24
Solve for x x :
x=6y+24y−1 x = \frac{6y + 24}{y - 1}
This expression gives us the inverse function, so:
f−1(x)=6x+24x−1 f^{-1}(x) = \frac{6x + 24}{x - 1}

STEP 5

Verify that f(f−1(x))=x f(f^{-1}(x)) = x :
Substitute f−1(x) f^{-1}(x) into f(x) f(x) :
f(f−1(x))=f(6x+24x−1) f(f^{-1}(x)) = f\left(\frac{6x + 24}{x - 1}\right)
=(6x+24x−1)+24(6x+24x−1)−6 = \frac{\left(\frac{6x + 24}{x - 1}\right) + 24}{\left(\frac{6x + 24}{x - 1}\right) - 6}
Simplify the expression:
Numerator:
6x+24+24(x−1)x−1=6x+24+24x−24x−1=30xx−1 \frac{6x + 24 + 24(x - 1)}{x - 1} = \frac{6x + 24 + 24x - 24}{x - 1} = \frac{30x}{x - 1}
Denominator:
6x+24−6(x−1)x−1=6x+24−6x+6x−1=30x−1 \frac{6x + 24 - 6(x - 1)}{x - 1} = \frac{6x + 24 - 6x + 6}{x - 1} = \frac{30}{x - 1}
f(f−1(x))=30xx−130x−1=x f(f^{-1}(x)) = \frac{\frac{30x}{x - 1}}{\frac{30}{x - 1}} = x

STEP 6

Verify that f−1(f(x))=x f^{-1}(f(x)) = x :
Substitute f(x) f(x) into f−1(x) f^{-1}(x) :
f−1(f(x))=f−1(x+24x−6) f^{-1}(f(x)) = f^{-1}\left(\frac{x + 24}{x - 6}\right)
=6(x+24x−6)+24(x+24x−6)−1 = \frac{6\left(\frac{x + 24}{x - 6}\right) + 24}{\left(\frac{x + 24}{x - 6}\right) - 1}
Simplify the expression:
Numerator:
6(x+24)+24(x−6)x−6=6x+144+24x−144x−6=30xx−6 \frac{6(x + 24) + 24(x - 6)}{x - 6} = \frac{6x + 144 + 24x - 144}{x - 6} = \frac{30x}{x - 6}
Denominator:
x+24−(x−6)x−6=30x−6 \frac{x + 24 - (x - 6)}{x - 6} = \frac{30}{x - 6}
f−1(f(x))=30xx−630x−6=x f^{-1}(f(x)) = \frac{\frac{30x}{x - 6}}{\frac{30}{x - 6}} = x

STEP 7

Choose the correct option for the inverse function:
The inverse function is valid for all x≠1 x \neq 1 because the denominator in f−1(x) f^{-1}(x) cannot be zero. Thus, the correct choice is:
A. f−1(x)=6x+24x−1 f^{-1}(x) = \frac{6x + 24}{x - 1} , for x≠1 x \neq 1
The inverse function is:
6x+24x−1 \boxed{\frac{6x + 24}{x - 1}}

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