Question

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.
{Use of Tech} $f$\left$(x$\right$)=$\frac{1}{x^2}$$

Accepted Answer

Step 1: Understand that for a function to have an inverse, it must be one-to-one (bijective). A function is one-to-one if it passes the horizontal line test, meaning no horizontal line intersects the graph of the function more than once. Step 2: Consider the function f(x) = \frac{1}{$$x^2$$}. Analyze its behavior: as x approaches 0 from either side, f(x) approaches infinity, and as x approaches positive or negative infinity, f(x) approaches 0. Step 3: Note that f(x) = \frac{1}{$$x^2$$} is not one-to-one over its entire domain because for any positive y, there are two x-values (one positive and one negative) that map to y. This violates the horizontal line test. Step 4: To find the largest possible set where f(x) has an inverse, restrict the domain to either x \u003e 0 or x \u003c 0. This restriction ensures that the function is one-to-one, as each y-value corresponds to exactly one x-value. Step 5: Conclude that the largest possible sets of points on which f(x) = \frac{1}{$$x^2$$} has an inverse are the intervals (0, ∞) or (-∞, 0). On these intervals, the function is strictly decreasing or increasing, respectively, and thus one-to-one.

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.
{Use of Tech} $f\(\left\)(x\(\right\))=\(\frac{1}{x^2}\)$

Verified video answer for a similar problem:

Key Concepts

Inverse Functions

Horizontal Line Test

Domain and Range

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.{Use of Tech} f(x)=1x2f\(\left\)(x\(\right\))=\(\frac{1}{x^2}\)f(x)=x21