Question

Determine which of these functions have an inverse function. Give a reason.

a. f(x) = x²
b. f(x) = 5x + 1
c. f(x) = 3
d. f(x)=√x
(This isn’t multiple choice, it’s asking for each one)

Answer 1

The function f(x) = x² does not have an inverse for all real numbers due to not being one-to-one. However, f(x) = 5x + 1 and f(x)=√x both have inverses as they pass the Horizontal Line Test. The constant function f(x) = 3 does not have an inverse.

Step-by-step explanation:

To determine whether each function has an inverse, we can use the Horizontal Line Test which states that if any horizontal line intersects the graph of a function at more than one point, then the function does not have an inverse because it is not one-to-one. Let's examine each function:

• f(x) = x². This function does not have an inverse over the entire real number line because it is not one-to-one; a horizontal line can intersect its graph in two points (e.g., y = 4 intersects at x = 2 and x = -2).
• f(x) = 5x + 1. This is a linear function with a non-zero slope, so it is one-to-one and therefore has an inverse function.
• f(x) = 3. This is a constant function and does not have an inverse because it is not one-to-one; a horizontal line will intersect its graph at every point along the line y = 3.
• f(x)=√x. This function does have an inverse, as it is one-to-one. Any horizontal line will intersect its graph at exactly one point.