Question

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.ƒ(x) = |2x + 1|

Accepted Answer

Step 1: Understand the concept of an inverse function. An inverse function exists if the original function is one-to-one (bijective), meaning it passes the horizontal line test. This implies that for every y-value, there is exactly one x-value. Step 2: Analyze the function $$ f(x) = |2x + 1| . $$The absolute value function is not one-to-one over its entire domain because it reflects negative inputs to positive outputs, causing multiple x-values to map to the same y-value. Step 3: Graphically, the function $$ f(x) = |2x + 1|  is a V-$$shaped graph with a vertex at the point where $$ 2x + 1 = 0 $$, which is $$ x = -\frac{1}{2} . $$The graph is symmetric about the vertical line $$ x = -\frac{1}{2} . $$Step 4: To find the largest possible set where the function has an inverse, consider restricting the domain to either side of the vertex. For example, restrict the domain to $$ x \geq -\frac{1}{2}  or$$ $$ x \leq -\frac{1}{2} . $$This restriction ensures the function is one-to-one on the chosen interval. Step 5: Verify the restricted function is one-to-one. For $$ x \geq -\frac{1}{2} $$, the function $$ f(x) = 2x + 1  is $$linear and increasing, thus one-to-one. Similarly, for $$ x \leq -\frac{1}{2} $$, the function $$ f(x) = -(2x + 1)  is $$linear and decreasing, also one-to-one.

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

ƒ(x) = |2x + 1|

Verified video answer for a similar problem:

Key Concepts

Inverse Functions

Horizontal Line Test

Piecewise Functions