Problem 24 - Calculus for Biology and Medicine

Question

(a) Graph \(f(x)=x, x \geq 0\), and \(g(x)=x^{2}, x \geq 0\), together, in one coordinate system. (b) For which values of \(x\) is \(f(x) \geq g(x)\), and for which values of \(x\) is \(f(x) \leq g(x)\) ?

Step-by-Step Solution

Answer

\(f(x) \geq g(x)\) for \(0 \leq x \leq 1\); \(f(x) \leq g(x)\) for \(x \geq 1\).

Step 1: Identify the Functions

We have two functions: \(f(x) = x\) and \(g(x) = x^2\). Both functions are defined for \(x \geq 0\). We'll graph these functions on the same coordinate plane.

Step 2: Graph the Functions

Firstly, plot the line \(f(x) = x\), which is a straight line through the origin with a slope of 1. Next, plot \(g(x) = x^2\), which is a parabola with its vertex at the origin that opens upwards. The graph of \(f(x) = x\) is linear, while \(g(x) = x^2\) is quadratic.

Step 3: Find Points of Intersection

To find where these functions intersect, set \(f(x) = g(x)\), which leads to \(x = x^2\). Solve the equation \(x^2 - x = 0\) by factoring: \(x(x-1) = 0\). It gives us solutions \(x = 0\) and \(x = 1\). These are the points where the two graphs intersect.

Step 4: Determine Sections of Inequality

Test intervals between intersection points to determine where \(f(x) = x\) is greater than or less than \(g(x) = x^2\). Choose test points in the intervals \([0, 1]\) and \([1, \infty)\).

Step 5: Check Interval \([0, 1]\)

Select a test point, for example, \(x = 0.5\). Calculate \(f(0.5) = 0.5\) and \(g(0.5) = (0.5)^2 = 0.25\). Since \(0.5 > 0.25\), in the interval \([0, 1]\), \(f(x) \geq g(x)\).

Step 6: Check Interval \([1, \infty)\)

Select a test point, for example, \(x = 2\). Calculate \(f(2) = 2\) and \(g(2) = 2^2 = 4\). Since \(2 < 4\), in the interval \([1, \infty)\), \(f(x) \leq g(x)\).

Key Concepts

Intersection of Functions

The intersection of two functions refers to the points where the graphs of these functions cross each other on a coordinate plane. To find the intersection point, we set the functions equal to each other. For example, given two functions, \( f(x) = x \) and \( g(x) = x^2 \), we solve the equation \( f(x) = g(x) \), which simplifies to \( x = x^2 \).
Factoring this equation, we have \( x(x - 1) = 0 \). This gives us the solutions \( x = 0 \) and \( x = 1 \), indicating the points where the two graphs intersect. These intersections are crucial as they reveal the points where both functions have the same output value. Understanding these intersections can help in analyzing the behavior of the functions and their comparative growth or decay within given domains.

Linear and Quadratic Functions

Linear functions, like \( f(x) = x \), represent straight lines on a graph. They have a constant rate of change and are defined by equations of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our case, the line \( f(x) = x \) crosses the origin, and its slope is \( 1 \).
Quadratic functions, like \( g(x) = x^2 \), form parabolas on a graph. These functions have a variable rate of change, represented by equations of the form \( y = ax^2 + bx + c \). The function \( g(x) = x^2 \) is a simple parabola with its vertex at the origin, opening upwards.
Both function types are fundamental in algebra, illustrating basic graph shapes and providing insights into real-world scenarios such as motion, growth, and optimization. Understanding their shapes and properties helps in predicting and interpreting mathematical and practical situations.

Inequalities in Functions

Inequalities in functions highlight the intervals where one function's output is greater than or less than another's. This is particularly useful when comparing rates of change or growth within the same graph. In our scenario, we examine the intervals where \( f(x) = x \) is greater than or equal to \( g(x) = x^2 \) and vice versa.

• For the interval \([0, 1]\), testing with a point like \( x = 0.5 \), we find \( f(0.5) = 0.5 \) and \( g(0.5) = 0.25 \). Here, \( f(x) \) is greater than \( g(x) \), so \( f(x) \geq g(x) \).
• For the interval \([1, \infty)\), using a test point such as \( x = 2 \), we calculate \( f(2) = 2 \) and \( g(2) = 4 \). This shows \( f(x) \leq g(x) \) since 2 is less than 4.

These intervals provide insight into function behaviors across different domains, further elucidating how outputs differ based on input values. This understanding is crucial in various mathematical applications from optimization problems to financial analysis.