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 step by step guidance

1. 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.
2. 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.
3. 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 \( x = -\frac{1}{2} \).
4. 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.
5. 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.

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(x) to have an inverse, it must be one-to-one, meaning that each output is produced by exactly one input. This property ensures that for every y-value in the range, there is a unique x-value in the domain, allowing us to 'solve' for x in terms of y.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, the function fails the test and does not have an inverse. This test is particularly useful for visualizing the behavior of functions and identifying intervals where they may be invertible.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of their domain. The function ƒ(x) = |2x + 1| is a piecewise function that can be expressed as two linear functions depending on the value of x. Understanding how to analyze these segments is crucial for determining where the function is one-to-one and thus where it has an inverse.