Question

inverse of 𝑓⁡(𝑥)=−2⁢𝑥+1. Graph the function and its inverse If necessary, restrict the domain of the inverse so that it is a function.

Answer 1

Find the inverse algebraically:

y = −2x + 1
Swap x and y: x = −2y + 1
Solve for y: −2y = x − 1 ⇒ y = (1 − x)/2

So f⁻¹(x) = (1 − x)/2 = 1/2 − (1/2)x.

Domain/range: f is one-to-one on all real numbers, so no restriction is needed; both f and f⁻¹ have domain and range = ℝ.

Graphing:
- f(x) = −2x + 1: slope −2, y-intercept (0,1). Points: (0,1), (1,−1), (−1,3).
- f⁻¹(x) = (1 − x)/2: slope −1/2, y-intercept (0,1/2). Points: (0,1/2), (1,0), (−1,1).

Note: f and f⁻¹ are reflections of each other across the line y = x; they intersect at the fixed point where f(x) = x, which is x = 1/3 (point (1/3,1/3)).