Problem 62 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=\sqrt{x^{2}+1}$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Recall Definitions of Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \), while an odd function satisfies \( f(-x) = -f(x) \) for all \( x \). We will use these definitions to determine the nature of the given function.

Step 2: Substitute \(-x\) into the Function

Substitute \(-x\) into the function \( f(x) = \sqrt{x^2+1} \), i.e., replace \( x \) with \( -x \). This gives us \( f(-x) = \sqrt{(-x)^2+1} \).

Step 3: Simplify the Expression

Since \((-x)^2 = x^2\), we can simplify \( f(-x) \) as follows: \( f(-x) = \sqrt{x^2 + 1} \).

Step 4: Compare \( f(-x) \) to \( f(x) \)

Compare the simplified expression for \( f(-x) \) with \( f(x) \). Notice that \( f(-x) = \sqrt{x^2+1} = f(x) \). Since \( f(-x) = f(x) \) for all \( x \), the function is even.

Key Concepts

Function Symmetry

When evaluating function symmetry, we need to decide whether a given function is even, odd, or neither. These concepts help in understanding how a function behaves when its input is negated. Symmetry can simplify our analysis and graphing of functions.

For even functions, we look for symmetry about the y-axis. This means that if you fold the graph along the y-axis, both halves would match up perfectly. Mathematically, an even function satisfies the equation \( f(-x) = f(x) \). This relationship implies that substituting \(-x\) for \(x\) won't change the value of \(f(x)\).

Odd functions, however, have a different type of symmetry. They are symmetric about the origin. If you rotate the graph of an odd function 180 degrees around the origin, it matches the original graph. Odd functions satisfy \( f(-x) = -f(x) \), which means flipping both x and y-axis changes values of \(f(x)\).

In our specific exercise with \(f(x) = \sqrt{x^2+1} \), checking symmetry involves substituting \(-x\) into the function and simplifying it to see if it's equal to \( f(x) \). We found out that the function is symmetric about the y-axis, thus confirming it's an even function.

Function Properties

Function properties give us insight into the nature and behavior of functions through mathematical relationships and expressions.

For even functions like \(f(x) = \sqrt{x^2 + 1}\), certain properties inherently follow. These types of functions are integral to various advanced mathematical concepts and can simplify complex problems. Here are important properties of even functions you should remember:

• They are symmetrical about the y-axis, meaning \(f(x) = f(-x)\) for all x in the function's domain.
• Presence of symmetry means that for any point \((a, b)\) on the graph, there's also a point \((-a, b)\).
• Even functions often produce duplicate intervals, leading to simplifying calculations and analytical methods.

In terms of applying these properties, even functions often show up in practical problems involving engineering and physics, where symmetry can be leveraged to streamline analysis.
Understanding these properties plays a pivotal role in predicting how a function will behave graphically and analytically, which feeds into our next section on graphical analysis.

Graphical Analysis

Graphical analysis helps us visualize and understand how a function's properties manifest on its graph.

With an even function like \(f(x) = \sqrt{x^2 + 1}\), the primary aspect to consider is its symmetry. Because it's an even function, its graph will be symmetrical about the y-axis. If you fold the graph down the y-axis, both sides, the positive and negative x, will mirror each other.

This particular function, \( \sqrt{x^2+1} \), is also always positive and starts at \(y = 1\) when \(x = 0\). This indicates that there is no intersection with the x-axis. The expression \( x^2+1 \) inside the square root is never zero or negative, which influences the function to remain above the x-axis.

We can expect this type of graph to rise gradually as x moves away from zero because the square root function grows larger as x does. But due to symmetry, whatever the graph looks like in the positive \(x\) domain, it will mirror itself on the negative \(x\) side.
By fully grasping the graphical representation of even functions, it makes it easier to predict behaviors without calculating every specific value. This understanding leads to insights that are especially valuable when operating within fields that require real-world applications or when performing extended computations, like calculus or algebra.