Problem 10 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

The function given is \(f(x) = x^2 - 7x\). This is a quadratic function with \(x\) being the variable.

Step 2: Substituting to Find \(f(a)\)

To find \(f(a)\), substitute \(a\) for \(x\) in \(f(x)\): \[f(a) = a^2 - 7a.\] Thus, the expression for \(f(a)\) is \(a^2 - 7a\).

Step 3: Substituting to Find \(f(a-3)\)

To find \(f(a-3)\), substitute \(a-3\) for \(x\) in \(f(x)\): \[f(a-3) = (a-3)^2 - 7(a-3).\] Simplify this expression: 1. Expand \((a-3)^2\) to get \(a^2 - 6a + 9\).2. Simplify \(-7(a-3)\) to \(-7a + 21\).3. Combine these results: \(a^2 - 6a + 9 - 7a + 21 = a^2 - 13a + 30\).Thus, \(f(a-3) = a^2 - 13a + 30\).

Step 4: Substituting to Find \(f(a+h)\)

To find \(f(a+h)\), substitute \(a+h\) for \(x\) in \(f(x)\): \[f(a+h) = (a+h)^2 - 7(a+h).\] Simplify this expression:1. Expand \((a+h)^2\) to get \(a^2 + 2ah + h^2\).2. Simplify \(-7(a+h)\) to \(-7a - 7h\).3. Combine these results: \(a^2 + 2ah + h^2 - 7a - 7h\).Thus, \(f(a+h) = a^2 + 2ah + h^2 - 7a - 7h\).

Key Concepts

Function Substitution

Function substitution is a method used to evaluate a function at specific values. It's especially useful in exploring relationships or patterns within functions. To achieve function substitution, you replace the variable in the function with the given value or expression.
In the context of our exercise, to find the value of the quadratic function \(f(x) = x^2 - 7x\) when \(x\) is replaced by another expression, we follow substitution steps:

• Identify the base function: in this case, \(f(x) = x^2 - 7x\).
• For \(f(a)\), substitute \(a\) for \(x\), giving \(f(a) = a^2 - 7a\).
• For \(f(a-3)\), substitute \(a-3\) for \(x\): \(f(a-3) = (a-3)^2 - 7(a-3)\).
• For \(f(a+h)\), substitute \(a+h\) for \(x\): \(f(a+h) = (a+h)^2 - 7(a+h)\).

By using substitution, you manipulate the function's structure to fit the new context or value, providing further insight into its behavior at those points.

Expansion of Binomials

The expansion of binomials is a key algebraic technique that involves multiplying out expressions of the form \((a+b)^2\) or \((a-b)^2\). The process involves applying distributive properties to eliminate parentheses and simplify the expressions.
In this exercise, expansion is needed when computing \(f(a-3)\) and \(f(a+h)\), as they involve expressions like \((a-3)^2\) and \((a+h)^2\):

• For \((a-3)^2\), distribute to get: \(a^2 - 6a + 9\).
• For \((a+h)^2\), distribute to get: \(a^2 + 2ah + h^2\).

Each expansion provides a rewritten form of the binomial, which can then be combined with other terms in the expression. Understanding this concept is crucial as it forms the foundation for solving many broader algebraic problems.

Algebraic Expressions

Algebraic expressions involve operations with variables and constants, resulting in expressions that can be simplified to explore their properties. These expressions combine constants, coefficients, and variables adhering to algebraic principles like associative, distributive, and commutative laws.
In our problem, we're dealing with algebraic manipulation while finding \(f(a-3)\) and \(f(a+h)\).

• Combine and simplify the expressions: After expansion, such as \((a-3)^2\) or \((a+h)^2\), combine similar terms to simplify the quadratic expressions.
• For \((a-3)^2 - 7(a-3)\), combine terms to get \(a^2 - 13a + 30\).
• For \((a+h)^2 - 7(a+h)\), combine terms to get \(a^2 + 2ah + h^2 - 7a - 7h\).

Handling algebraic expressions with ease helps in simplifying complicated polynomial expressions, making it a critical skill in solving mathematical problems.