Question

Find and graph the inverse function (x) of the piecewise function defined by: 2x + 3 & \text{if } x \leq 0 \\ x^2 + 3 & \text{if } x > 0 \end{cases}

Answer 1

To find the inverse function of a piecewise function, solve for x in each piece individually. For the first piece, 2x + 3, the inverse is (y - 3) / 2. For the second piece, x^2 + 3, the inverse is sqrt(y - 3). Combine these inverses based on the respective domains, resulting in the overall inverse function.

Step-by-step explanation:

To determine the inverse function of a piecewise function, first find the inverse of each piece separately. For the first piece, 2x + 3, solve for x, yielding (y - 3) / 2.

For the second piece, x^2 + 3, solve for x, giving sqrt(y - 3).

Combine these inverses based on their respective domains: the first piece for x ≤ 0 and the second for x > 0.

The resulting inverse function comprises two pieces: (y - 3) / 2 when y ≤ 3 and sqrt(y - 3) when y > 3.

This represents the complete inverse function for the given piecewise function, accommodating different domains for each piece.

Find and graph the inverse function f −1(x) of the piecewise function defined by

f(x)= { 2x + 3 if x ≤ 0

x2+3 if0<x≤2

log2(x−1)+7 ifx>2.