Math  /  Algebra

Questionf(x)=3x12x,f1(x)=?f(x)=\frac{3 x}{1-2 x}, \quad f^{-1}(x)=?

Studdy Solution

STEP 1

1. We are given the function f(x)=3x12x f(x) = \frac{3x}{1-2x} .
2. We need to find the inverse function f1(x) f^{-1}(x) .

STEP 2

1. Set y=f(x) y = f(x) to express the function in terms of y y .
2. Solve the equation for x x in terms of y y .
3. Swap x x and y y to find the inverse function.
4. Verify the inverse function by checking if f(f1(x))=x f(f^{-1}(x)) = x .

STEP 3

Set y=f(x) y = f(x) :
y=3x12x y = \frac{3x}{1-2x}

STEP 4

Solve for x x in terms of y y . Start by multiplying both sides by 12x 1-2x to eliminate the fraction:
y(12x)=3x y(1-2x) = 3x

STEP 5

Distribute y y on the left side:
y2yx=3x y - 2yx = 3x

STEP 6

Rearrange terms to isolate terms involving x x on one side:
y=3x+2yx y = 3x + 2yx

STEP 7

Factor out x x from the right side:
y=x(3+2y) y = x(3 + 2y)

STEP 8

Solve for x x :
x=y3+2y x = \frac{y}{3 + 2y}

STEP 9

Swap x x and y y to find the inverse function:
f1(x)=x3+2x f^{-1}(x) = \frac{x}{3 + 2x}

STEP 10

Verify the inverse by checking if f(f1(x))=x f(f^{-1}(x)) = x :
Substitute f1(x) f^{-1}(x) into f(x) f(x) :
f(x3+2x)=3(x3+2x)12(x3+2x) f\left(\frac{x}{3 + 2x}\right) = \frac{3\left(\frac{x}{3 + 2x}\right)}{1 - 2\left(\frac{x}{3 + 2x}\right)}
Simplify the expression:
=3x3+2x3+2x2x3+2x = \frac{\frac{3x}{3 + 2x}}{\frac{3 + 2x - 2x}{3 + 2x}} =3x3 = \frac{3x}{3} =x = x
Since f(f1(x))=x f(f^{-1}(x)) = x , the inverse function is verified.
The inverse function is:
f1(x)=x3+2x f^{-1}(x) = \frac{x}{3 + 2x}

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