Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{4}-7 x^{2}+6$$

Step-by-Step Solution

Answer

The function \(f(x) = x^{4} - 7x^{2} + 6\) is even.

Step 1: Understand the Definitions

A function is even if replacing every occurrence of \(x\) with \(-x\) results in the original function, \(f(-x) = f(x)\). A function is odd if replacing every occurrence of \(x\) with \(-x\) results in the negative of the original function, \(f(-x) = -f(x)\). We need to evaluate \(f(-x)\) and compare it with the original function \(f(x)\).

Step 2: Substitute -x into f(x)

Calculate \(f(-x)\) by substituting \(-x\) into \(f(x)=x^{4}-7x^{2}+6\):\[f(-x) = (-x)^{4} - 7(-x)^{2} + 6\]

Step 3: Simplify the Expression

Simplify each part of \(f(-x)\):1. \((-x)^4 = x^4\) because raising to an even power negates the negative sign.2. \((-x)^2 = x^2\), also a result due to squaring the negative.Thus, the expression simplifies to:\[f(-x) = x^4 - 7x^2 + 6\]

Step 4: Compare with f(x)

Compare \(f(-x)\) and \(f(x)\):\(f(-x) = x^4 - 7x^2 + 6\) is equal to \(f(x) = x^4 - 7x^2 + 6\).Since \(f(-x) = f(x)\), the function is even.

Key Concepts

Understanding Function Symmetry

Function symmetry is a fascinating concept that helps us understand the geometric properties of a graph. It is crucial for determining whether a function is even, odd, or neither. Here's the idea in a nutshell:

• An even function has symmetry with respect to the y-axis. This means if you were to fold the graph along the y-axis, the two halves would match exactly. Mathematically, an even function satisfies the condition: \(f(-x) = f(x)\).
• An odd function exhibits rotational symmetry around the origin. This is achieved when the graph looks the same after a rotation of 180 degrees about the origin. For an odd function, the condition is: \(f(-x) = -f(x)\).

In the exercise, evaluating the function \(f(x) = x^4 - 7x^2 + 6\) tells us whether it is even or odd by examining symmetry properties.Understanding these conditions allows us to infer much about the function's structure and behavior, which is key in calculus and graphing.

The Process of Function Evaluation

Function evaluation involves substituting given inputs into a function to see how it behaves. For example, the task is to input \(-x\) into the function \(f(x)\) and simplify the resulting expression. Let's break it down:

• Substitute \(-x\) in place of \(x\) in the function. For our function \(f(x) = x^4 - 7x^2 + 6\), this becomes: \[f(-x) = (-x)^4 - 7(-x)^2 + 6\]
• Simplify the expression. Remember that when we raise a negative number to an even power, the result is positive. So, \((-x)^4 = x^4\) and \((-x)^2 = x^2\). Hence, \[f(-x) = x^4 - 7x^2 + 6\]

By comparing this with the original function, you can confirm whether the function is even or odd. Function evaluation is handy not only for verifying symmetry but also for determining function output at specific inputs, solving equations, and more.

Exploring Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are foundational in algebra and calculus, and understanding them is key to mastering these subjects. At the heart of the exercise is the polynomial function \(f(x) = x^4 - 7x^2 + 6\). Let's look at its components:

• The term \(x^4\) is a polynomial term of degree 4, as the highest power of \(x\) is 4.
• The term \(-7x^2\) is a second-degree polynomial term (quadratic term), contributing significantly to the function's shape.
• The constant \(+6\) represents a vertical shift in the graph of the function.

These characteristics influence the graph's behavior, such as its curvature and the number and nature of turning points. Polynomial functions are generally smooth and continuous, making them predictable and easier to manipulate when solving for roots or evaluating limits. In higher mathematics, the symmetry of these polynomials often aids in finding solutions and making predictions about function behavior.