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) \(f(x)=\frac{1}{x}, x>0\) (d) \(f(x)=e^{x}, x \in \mathbf{R}\) (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), (d), and (f) are one-to-one.

Step 1: Understand the One-to-One Function

A function is one-to-one if every horizontal line cuts the graph at most once. With the horizontal line test, if any horizontal line intersects the graph more than once, the function is not one-to-one.

Step 2: Examine Function (a)

Consider the function \(f(x) = x^2\) where \(x \geq 0\). Graphically, this function represents a parabola opening upwards starting from the origin. In this domain constraint, any horizontal line will intersect the graph at most once, thus this function is one-to-one on \(x \geq 0\).

Step 3: Examine Function (b)

For \(f(x) = x^2\) where \(x \in \mathbf{R}\), the entire real set, the graph is a symmetric parabola opening upwards, intersecting any horizontal line at two points unless the horizontal line is the vertex. Therefore, the function is not one-to-one when \(x \in \mathbf{R}\).

Step 4: Examine Function (c)

Consider \(f(x) = \frac{1}{x}\) where \(x > 0\). This graph forms a hyperbola in the first quadrant, asymptotic to both axes, and each horizontal line intersects the graph only once, implying the function is one-to-one for \(x > 0\).

Step 5: Examine Function (d)

The function \(f(x) = e^{x}\) is an exponential graph that continuously rises as \(x\) increases. Any horizontal line will intersect this graph at most once, confirming \(f(x) = e^{x}\) is one-to-one for \(x \in \mathbf{R}\).

Step 6: Examine Function (e)

For \(f(x) = \frac{1}{x^2}\) when \(x eq 0\), the graph forms symmetric but separate hyperbolic curves in the first and third quadrants. Horizontal lines can intersect both sections of the graph simultaneously, showing this function is not one-to-one.

Step 7: Examine Function (f)

Consider \(f(x) = \frac{1}{x^2}\) where \(x > 0\). This graph is a hyperbola only in the first quadrant. However, any horizontal line intersects it in exactly one place as the graph decreases to the right without lowering under x-axis, showing it's one-to-one for the given domain.

Key Concepts

Horizontal Line Test

To determine if a function is one-to-one, you can perform the horizontal line test. This simple yet powerful test involves drawing horizontal lines across the graph of a function to see how many times the line intersects the graph. If any horizontal line intersects the graph more than once, the function is not one-to-one. This test helps us quickly analyze the behavior of a function without having to delve into complex calculations.

• If a function passes the horizontal line test, it means that for every output, there is exactly one input.
• A graph that passes the test is important in determining whether a function has an inverse function.

This test can be used effectively for a range of functions including linear, quadratic, exponential, and rational functions. For instance, in the exercise, the function \( f(x) = x^2 \) with \( x \geq 0 \), passes the horizontal line test because every horizontal line intersects it at most once due to its restricted domain.

Function Analysis

When analyzing a function to determine its properties, it's crucial to consider how the function behaves in terms of its graph and algebraic expression. Function analysis involves looking at details like the function's domain, range, and any restrictions given to the variables. It gives insight into whether or not a function is one-to-one.
For function \( f(x) = x^2 \) with \( x \in \mathbb{R} \), being a symmetric parabola shows that it's not one-to-one, as any horizontal line above the vertex will intersect the graph twice. Meanwhile, restricting the domain to \( x \geq 0 \), transforms this function into one that is one-to-one since it only allows the right half of the parabola to show, avoiding repeated \( y \)-values for different \( x \)-values.

• Domain plays a key role in determining whether functions like \( f(x) = \frac{1}{x^2} \) or \( f(x) = e^x \) are one-to-one.
• Understanding both the graphical and algebraic properties provides a comprehensive view of the function's behavior.

Calculus Concepts

Understanding calculus concepts can further enhance our analysis of functions. Calculus introduces tools like derivatives that help determine properties such as monotonicity – identifying whether a function is consistently increasing or decreasing. A monotonically increasing or decreasing function is naturally one-to-one.
Consider the function \( f(x) = e^x \); it is always increasing because its derivative \( f'(x) = e^x \) is always positive. Therefore, any horizontal line can only touch the graph once, passing the horizontal line test.

• Monotonic functions, whether they continuously increase or decrease, simplify the study of one-to-one functions.
• The derivative function provides valuable information on intervals of increase/decrease.

Hence, calculus not only helps in assessing one-to-one functions but also aids in understanding broader aspects such as curves' behaviors and tangent slopes, which further support effective function analysis.