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Math  /  Algebra

QuestionLet f(x)=−3x+4f(x)=-3 x+4. Then f−1(x)=f^{-1}(x)=

Studdy Solution

STEP 1

1. We are given a linear function f(x)=−3x+4 f(x) = -3x + 4 .
2. We need to find the inverse function, denoted as f−1(x) f^{-1}(x) .
3. The function f(x) f(x) is one-to-one, which means it has an inverse.

STEP 2

1. Understand the concept of an inverse function.
2. Set up the equation to find the inverse.
3. Solve for the inverse function.
4. Verify the inverse function.

STEP 3

An inverse function, f−1(x) f^{-1}(x) , essentially reverses the operation of the original function f(x) f(x) . If f(a)=b f(a) = b , then f−1(b)=a f^{-1}(b) = a .

STEP 4

To find the inverse, start by replacing f(x) f(x) with y y :
y=−3x+4 y = -3x + 4
Next, swap x x and y y to set up the equation for the inverse function:
x=−3y+4 x = -3y + 4

STEP 5

Solve for y y to find the inverse function:
First, subtract 4 from both sides:
x−4=−3y x - 4 = -3y
Then, divide both sides by −3-3:
y=x−4−3 y = \frac{x - 4}{-3}
Simplify the expression:
y=−x3+43 y = -\frac{x}{3} + \frac{4}{3}
Thus, the inverse function is:
f−1(x)=−x3+43 f^{-1}(x) = -\frac{x}{3} + \frac{4}{3}

STEP 6

Verify the inverse function by checking if f(f−1(x))=x f(f^{-1}(x)) = x and f−1(f(x))=x f^{-1}(f(x)) = x .
First, check f(f−1(x)) f(f^{-1}(x)) :
f(−x3+43)=−3(−x3+43)+4 f\left(-\frac{x}{3} + \frac{4}{3}\right) = -3\left(-\frac{x}{3} + \frac{4}{3}\right) + 4
Simplify:
=x−4+4=x = x - 4 + 4 = x
Now, check f−1(f(x)) f^{-1}(f(x)) :
f−1(−3x+4)=−−3x+43+43 f^{-1}(-3x + 4) = -\frac{-3x + 4}{3} + \frac{4}{3}
Simplify:
=x = x
Both conditions are satisfied, confirming that the inverse function is correct.

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