Problem 69 - Calculus for Biology and Medicine

Question

Which of the following functions is one to one (use the horizontal line test)? (a) \(f(x)=x^{2}, x \geq 0\) (b) \(f(x)=x^{2}, x \in \mathbf{R}\) (c) \(f(x)=\sqrt{x}, x \geq 0\) (d) \(f(x)=\ln x, x>0\) (e) \(f(x)=\frac{1}{x^{2}}, x \neq 0\) (f) \(f(x)=\frac{1}{x^{2}}, x>0\)

Step-by-Step Solution

Answer

Functions (a), (c), and (d) are one-to-one.

Step 1: Understanding One-to-One Functions

A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. This is known as the horizontal line test.

Step 2: Analyze Function (a)

For function (a) \(f(x)=x^2, x \geq 0\), the horizontal line test shows that each horizontal line will intersect the function at most once because the domain is restricted to non-negative \(x\) values. This makes it one-to-one on this domain.

Step 3: Analyze Function (b)

For function (b) \(f(x)=x^2, x \in \mathbf{R}\), horizontal lines will intersect the graph at more than one point because the parabola extends to negative \(x\) values as well. Thus, it is not one-to-one.

Step 4: Analyze Function (c)

For function (c) \(f(x)=\sqrt{x}, x \geq 0\), every horizontal line in the graph intersects at most once, so this function is one-to-one.

Step 5: Analyze Function (d)

For function (d) \(f(x)=\ln x, x>0\), the curve never repeats its value as \(x\) increases, meaning every horizontal line crosses it only once. This function is one-to-one.

Step 6: Analyze Function (e)

For function (e) \(f(x)=\frac{1}{x^2}, x eq 0\), the graph is symmetrical about the y-axis and horizontal lines intersect the graph at more than one point on either side of the y-axis. This function is not one-to-one.

Step 7: Analyze Function (f)

For function (f) \(f(x)=\frac{1}{x^2}, x>0\), the graph is only on the positive \(x\) side, but horizontal lines still intersect it more than once as \(x\) approaches zero and infinity, meaning the function is not one-to-one.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple and powerful tool used to determine whether a function is one-to-one. This test involves drawing horizontal lines (parallel to the x-axis) across the graph of the function.

• If every horizontal line intersects the function's graph at most once, then the function is one-to-one.
• If any horizontal line intersects the graph more than once, the function is not one-to-one.

This test is useful because a one-to-one function has a unique output for every input and a unique input for every output. This characteristic is essential when considering the possibility of an inverse function.

Function Analysis

Analyzing a function involves understanding its behavior, such as where it is increasing or decreasing. These characteristics are crucial in determining whether a function is one-to-one.
For example, a function like a) \( f(x) = x^2, x \geq 0 \) is one-to-one in its restricted domain because as you move along the graph from left to right, each horizontal line intersects the graph only once.
Contrast this with function b) \( f(x) = x^2, x \in \mathbf{R} \). Here, the negative and positive values of x both produce the same square value, meaning horizontal lines will intersect the graph twice. Hence, function b is not one-to-one.
Carefully analyzing the section of the graph included in the domain helps clarify whether a function is one-to-one.

Domain Restrictions

Sometimes, a function can be altered to be one-to-one by restricting its domain. This is particularly common in functions like polynomials, where different portions of the graph behave differently.

• For instance, the function \( f(x) = x^2 \) is not one-to-one over all real numbers, because both positive and negative values of x yield the same y value.
• By restricting the domain to \( x \geq 0 \), the function becomes one-to-one because now, each horizontal line will intersect the function's graph only once.

Domain restrictions are thus a valuable technique to ensure that a function can pass the horizontal line test, especially when preparing to find an inverse function.

Inverse Functions

Inverse functions reverse the roles of inputs and outputs such that the operation of the inverse function returns the original input.

• For a function to have an inverse, it must be one-to-one. This ensures that each output is uniquely traceable back to one input, making the inverse function well-defined.
• When a function is one-to-one, we can create its inverse function by swapping the x and y variables and solving for the new output variable.

An excellent example of this is the natural logarithm function \( f(x) = \ln x, x > 0 \), which is one-to-one because it passes the horizontal line test. The inverse of the logarithm function is the exponential function. Thus, understanding if a function is one-to-one is the first step in determining if it has an inverse.