Problem 98 - College Algebra Essentials

Question

Find \(f(-x)-f(x)\) for the given function \(f\) Then simplify the expression. $$ f(x)=x^{2}-3 x+7 $$

Step-by-Step Solution

Answer

The simplified expression for \(f(-x) - f(x)\) is \(6x\).

Step 1: Find \(f(-x)\)

To find \(f(-x)\), replace every \(x\) in the function with \(-x\). So, \(f(-x) = (-x)^{2}-3(-x)+7\). Simplifying this, we get \(f(-x)=x^{2}+3x+7\)

Step 2: Find \(f(x)\)

The given function is \(f(x) = x^{2} - 3x + 7\)

Step 3: Calculate \(f(-x) - f(x)\)

Now subtract \(f(x)\) from \(f(-x)\) to obtain \(f(-x) - f(x) = (x^{2} + 3x + 7) - (x^{2} - 3x + 7)\). Simplifying this, we get \(f(-x)-f(x)=6x\)

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression consisting of variables raised to whole number exponents and combined using addition, subtraction, or multiplication. The general form of a polynomial function is given by:

• \( f(x) = a_n \, x^n + a_{n-1} \, x^{n-1} + \dots + a_1 \, x + a_0 \)

where the \( a_i\) are constants (coefficients), and \( n \) is a non-negative integer that represents the highest power of \( x \) (called the degree of the polynomial). Polynomials are fundamental in algebra because they are used to construct polynomial functions, like the one given in the problem: \( f(x) = x^2 - 3x + 7 \).
In this function:

• The highest degree term is \( x^2 \), making it a quadratic polynomial.
• \(-3x \) is the linear term, representing the first degree of \( x \).
• 7 is the constant term with no variable attached.

Understanding each part of the polynomial helps to recognize patterns, simplify expressions, and perform algebraic operations.

Function Operations

Function operations allow us to manipulate and combine functions in various ways. One important operation is the substitution you perform when evaluating a function at different points. In the exercise, different substitutions were performed:

• Substituting \(-x \) for \(x \) to find \( f(-x) \): Replacing \(x\) with \(-x\) gives \( f(-x) = (-x)^2 - 3(-x) + 7 = x^2 + 3x + 7 \).
• Subtracting \( f(x) \): Finding \( f(-x) - f(x) \) involves subtracting the original polynomial from its altered form.

This subtraction showcases a function operation where you explicitly compare the behavior of the function at opposite values of \(x\). It's useful in examining symmetry and other properties.
Other common operations include:

• Addition: \((f+g)(x) = f(x) + g(x)\)
• Multiplication: \((f\cdot g)(x) = f(x) \times g(x)\)
• Division: \(\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \) (with \(g(x) eq 0\))

These operations are fundamental in calculus and higher-level math, allowing us to build more complex models from simpler functions.

Symmetry in Algebra

Symmetry in algebra often refers to a function being even, odd, or possessing some symmetrical characteristics when graphed on a Cartesian plane. Using algebraic manipulation, we can determine the symmetry type of a function.
To check for symmetry:

• An even function satisfies \( f(-x) = f(x) \) for all \( x \). Its graph is symmetric about the y-axis.
• An odd function satisfies \( f(-x) = -f(x) \) for all \( x \). Its graph is symmetric about the origin.

In the exercise, you were asked to compute \( f(-x) - f(x) \). The result, \( 6x \), suggests that the given function is neither even nor odd. Instead, it shows a particular kind of symmetry that is associated with linear terms.
Understanding symmetry is crucial because it helps in predicting behavior without detailed calculations, simplifies computations, and aids in graph sketching.
Make sure to test different cases of \(x\), especially positive or negative values, to see these properties in action. Symmetry in polynomial functions allows us to conclude more with less computation, making problem-solving more efficient.