Problem 73 - Calculus

Question

Give an example of: An even function whose graph does not contain the point (0,0).

Step-by-Step Solution

Answer

An example is \(f(x) = x^2 + 2\), which is even and does not pass through (0,0).

Step 1: Understanding the Definition of an Even Function

An even function satisfies the property that for every input \(x\), \( f(x) = f(-x) \). In simpler terms, the function is symmetric with respect to the y-axis. An even function means that if you know one side of the function, the other side is a mirror reflection.

Step 2: Analyzing the Graph's Requirements

The problem specifies that the even function's graph should not pass through the point \((0,0)\). This means that \(f(0)\) should not be equal to zero. Therefore, we need to choose an equation for \(f(x)\) such that when \(x = 0\), \(f(0) eq 0\).

Step 3: Choosing a Suitable Even Function

A common even function is \( f(x) = x^2 \), but it passes through \((0,0)\). Instead, let's adjust this to avoid using zero: consider the function \(f(x) = x^2 + 2\).

Step 4: Verifying the Properties of the Chosen Even Function

Check that \(f(x) = x^2 + 2\) is indeed an even function. For every \(x\), we should have \(f(x) = f(-x)\). Calculating this, \( f(x) = x^2 + 2 = (-x)^2 + 2 = f(-x) \), confirming the function is even. Also, \( f(0) = 0^2 + 2 = 2 \), confirming the graph does not pass through \( (0,0) \).

Key Concepts

Function Symmetry

Function symmetry is a crucial concept in understanding how functions behave. A function is said to exhibit symmetry if its graph remains unchanged under certain transformations or reflections. Specifically, for even functions, the symmetry is with respect to the y-axis. This means that the left half of the graph is a mirror image of the right half.

In mathematical terms, an even function is one that satisfies the equation \( f(x) = f(-x) \) for all values of \( x \) in the function's domain. This property means that if you know the value of the function at any positive \( x \), you automatically know its value at \( -x \), since they are the same.

This concept is particularly useful when analyzing graphs, as symmetry can simplify calculations and help in visualizing the function's behavior.

Function Graph

The graph of a function provides a visual representation of how the function behaves over its domain. For an even function, the graph has a symmetric shape with respect to the y-axis. In this context, symmetry means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\) as well.

When you are plotting an even function, like \( f(x) = x^2 + 2 \), the graph will appear the same on both sides of the y-axis, exhibiting a mirrored pattern. This helps in quickly recognizing the nature of the function when glancing at its graph.

Graphing an even function is an excellent way to verify its properties, as you can visually check the symmetry around the y-axis, ensuring the function behaves as expected.

Y-axis Symmetry

Y-axis symmetry is a specific type of symmetry important when studying functions. A function displays y-axis symmetry if flipping the graph over the y-axis results in the same image.

In mathematical terms, this means that \( f(x) = f(-x) \) for all \( x \) in its domain. This feature is fundamental for even functions. It ensures that for each positive x-value, there is a corresponding negative x-value with the same function output.

Understanding y-axis symmetry helps simplify problems in calculus and algebra by reducing the need to analyze both sides of the graph separately. You only need to study one side, and you automatically know the behavior of the other due to this symmetry.

Function Properties

Functions come with various properties that help define their behavior and characteristics. For even functions, the main property is their symmetry about the y-axis, expressed as \( f(x) = f(-x) \). This property implies specific behaviors such as constant y-values for opposite x-values.

Analyzing function properties helps in identifying key features such as intercepts, maxima, minima, and intervals of increase or decrease. For an even function like \( f(x) = x^2 + 2 \), significant properties include:

• Domain: All real numbers, since no square root or division by zero issues exist.
• Range: Values starting from 2 upwards, since \( x^2 \) is always non-negative, making the smallest value \( f(x) = 2 \).
• Intercepts: The y-intercept is the value of the function at \( x=0 \), here it's 2, and there are no x-intercepts since \( x^2 + 2 = 0 \) has no real solutions.

Understanding these properties allows you to gain a complete picture of the function’s behavior, making it easier to predict and analyze its graph.