Problem 12 - Algebra for College Students

Question

If \(f(x)=2 x^{2}-x-1\), find \(f(-a), f(a+1)\), and \(f(a+h)\).

Step-by-Step Solution

Answer

\(f(-a) = 2a^2 + a - 1\), \(f(a+1) = 2a^2 + 3a + 1\), \(f(a+h) = 2a^2 + 4ah + 2h^2 - a - h - 1\).

Step 1: Understand the Function

The function given is a quadratic function: \(f(x) = 2x^2 - x - 1\). We'll evaluate this function at different inputs: \(-a\), \(a+1\), and \(a+h\).

Step 2: Calculate \(f(-a)\)

Substitute \(-a\) for \(x\) in the function: \[f(-a) = 2(-a)^2 - (-a) - 1\]Evaluate each term: - \((-a)^2 = a^2\), so \(2(-a)^2 = 2a^2\).- \(-(-a) = a\).Plug these into the function:\[f(-a) = 2a^2 + a - 1\]

Step 3: Calculate \(f(a+1)\)

Substitute \(a+1\) for \(x\):\[f(a+1) = 2(a+1)^2 - (a+1) - 1\]Use the binomial expansion to find \((a+1)^2 = a^2 + 2a + 1\): Thus,\[2(a+1)^2 = 2(a^2 + 2a + 1) = 2a^2 + 4a + 2\]Now substitute back into the function:\[f(a+1) = 2a^2 + 4a + 2 - a - 1 = 2a^2 + 3a + 1\]

Step 4: Calculate \(f(a+h)\)

Substitute \(a+h\) for \(x\):\[f(a+h) = 2(a+h)^2 - (a+h) - 1\]Use the binomial expansion for \((a+h)^2 = a^2 + 2ah + h^2\):\[2(a+h)^2 = 2(a^2 + 2ah + h^2) = 2a^2 + 4ah + 2h^2\]Substitute back to find the function:\[f(a+h) = 2a^2 + 4ah + 2h^2 - a - h - 1\]

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for specific input values. It involves substituting a given value for the variable in the function and simplifying to find the result. To evaluate a function like the quadratic function given, such as \( f(x) = 2x^2 - x - 1 \), you replace the input \( x \) with the specified value. For example, if the input is \(-a\), you substitute \(-a\) wherever you see \( x \). This requires careful substitution to ensure accuracy:

• Replace each occurrence of \(x\) with \(-a\), \(a+1\), or \(a+h\) as directed.
• Simplify the expression step by step, adhering to arithmetic rules.

This procedure allows you to determine specific outputs of quadratic functions, which can represent real-world phenomena like projectile motion.

Binomial Expansion

The binomial expansion refers to the algebraic process used to expand expressions raised to a power, specifically those of the form \((a + b)^n\). In the context of this exercise, binomial expansion is essential when evaluating a function at non-standard inputs like \((a+1)\) or \((a+h)\).For instance, when calculating \( f(a+1) \), we need to expand \((a+1)^2\):

• Recognize the pattern: \((a + 1)^2 = a^2 + 2a + 1\).
• This step simplifies complex polynomial terms, allowing them to be multiplied by other components easily.

By consistently applying the binomial expansion, we can unravel more complex expressions into manageable, calculable parts, leading to quicker and clearer function evaluation.

Polynomial Expressions

Polynomial expressions are mathematical expressions that include variables raised to whole number powers and multiplied by coefficients, usually combined through addition or subtraction. A quadratic function, such as \( f(x) = 2x^2 - x - 1 \), is a specific type of polynomial expression.Understanding polynomial expressions involves knowing how they are structured:

• The degree of a polynomial is determined by the highest power of the variable (e.g., the degree is 2 for quadratic polynomials).
• Each term consists of a coefficient and a variable raised to a power.

The manipulation of polynomial expressions, including expanding, simplifying, and evaluating, is fundamental in mathematics, especially in algebra and calculus, where such expressions frequently arise in various functions and equations.