Problem 49 - Intermediate Algebra

Question

If \(f(x)=x^{2}-7 x\), find \(f(a), f(a-3)\), and \(f(a+h)\).

Step-by-Step Solution

Answer

\( f(a) = a^2 - 7a \), \( f(a-3) = a^2 - 13a + 30 \), \( f(a+h) = a^2 - 7a + 2ah + h^2 - 7h \).

Step 1: Find \( f(a) \)

To find \( f(a) \), substitute \( a \) for \( x \) in the function \( f(x) = x^2 - 7x \). This gives, \[ f(a) = a^2 - 7a \].

Step 2: Find \( f(a-3) \)

Substitute \( a-3 \) for \( x \) in the function \( f(x) = x^2 - 7x \).This gives,\[ f(a-3) = (a-3)^2 - 7(a-3) \].Simplify the expression:1. Expand \( (a-3)^2 \): \[ (a-3)^2 = a^2 - 6a + 9 \].2. Expand \( -7(a-3) \): \[ -7(a-3) = -7a + 21 \].3. Combine the terms: \[ f(a-3) = a^2 - 6a + 9 - 7a + 21 \]. \[ f(a-3) = a^2 - 13a + 30 \].

Step 3: Find \( f(a+h) \)

Substitute \( a+h \) for \( x \) in the function \( f(x) = x^2 - 7x \).This gives,\[ f(a+h) = (a+h)^2 - 7(a+h) \].Simplify the expression:1. Expand \( (a+h)^2 \): \[ (a+h)^2 = a^2 + 2ah + h^2 \].2. Expand \( -7(a+h) \): \[ -7(a+h) = -7a - 7h \].3. Combine the terms: \[ f(a+h) = a^2 + 2ah + h^2 - 7a - 7h \]. \[ f(a+h) = a^2 - 7a + 2ah + h^2 - 7h \].

Key Concepts

Polynomial Functions

A polynomial function is an expression that involves multiple terms. Each term is a product of a constant and a variable raised to a non-negative integer power. In the given exercise, the function \(f(x) = x^2 - 7x\) is a simple polynomial function. It's known as a quadratic function because the highest power of the variable \(x\) is 2. Polynomials can have coefficients, which are the numbers in front of the variables. In this case, the polynomial has a coefficient of 1 for \(x^2\) and a coefficient of -7 for \(x\).
Understanding polynomial functions is fundamental in algebra. They are used to represent a wide range of real-world situations. When dealing with polynomials, expect to see different operations like addition, subtraction, multiplication, and division applied to the terms within the polynomial.
Important points about polynomial functions include:

• They have a finite number of terms.
• Each term consists of a coefficient and a variable raised to an exponent.
• The exponents in polynomial functions are non-negative integers.

Substitution Method

The substitution method is a technique used to evaluate functions by replacing the variable with a specific value or expression. In the given exercise, you are asked to find \(f(a)\), \(f(a-3)\), and \(f(a+h)\) by substituting different expressions for the variable \(x\).
Applying the substitution method is straightforward. Follow these steps:

• Identify the expression or value to substitute for the variable.
• Replace the occurrence of the variable in the function with the given expression or value.
• Simplify the resulting expression if needed.

For example, to find \(f(a-3)\), substitute \(a-3\) for \(x\) in \(f(x) = x^2 - 7x\), and then simplify the expression. This method is a powerful tool in algebra to explore how a function behaves with different inputs.
Using the substitution method often requires careful algebraic manipulation, such as expanding and simplifying expressions to express the function in a clearer form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the backbone of algebra and are essential for understanding mathematical functions, such as polynomial functions. In our exercise, expressions like \(a^2 - 7a\), \(a^2 - 13a + 30\), and \(a^2 - 7a + 2ah + h^2 - 7h\) are all examples of algebraic expressions.
Understanding algebraic expressions involves recognizing:

• Variables, which are symbols that represent unknown values.
• Coefficients, which are numbers multiplied by the variables.
• Constants, which are fixed values without variables.
• The use of arithmetic operations like addition, subtraction, multiplication, and division.

One crucial aspect of working with algebraic expressions is the simplification process. Simplifying involves combining like terms, which are terms with the same variables raised to the same power. For example, when simplifying \(f(a-3)\), you'll combine terms with the variable \(a\) to achieve the simplest form \(a^2 - 13a + 30\).
Algebraic expressions are versatile and appear in various mathematical contexts, providing a framework for problem-solving and logical reasoning.