Problem 43 - Intermediate Algebra

Question

Find each function value if \(f(x)=3 x^{2}-2 .\) See Section 3.2 $$ f(a) $$

Step-by-Step Solution

Answer

The function value is \(f(a) = 3a^2 - 2\).

Step 1: Understand What is Asked

You are asked to evaluate the function \(f(x) = 3x^2 - 2\) at \(x = a\). This means you need to substitute \(a\) into the function in place of \(x\).

Step 2: Substitute \(a\) into the Function

Replace \(x\) in the function \(f(x) = 3x^2 - 2\) with \(a\). This gives us the expression: \(f(a) = 3a^2 - 2\).

Step 3: Write Down the Final Function Value

The function value is now expressed as \(f(a) = 3a^2 - 2\). This means that the value of the function at \(x = a\) is \(3a^2 - 2\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, letters such as \(x\), \(y\), or \(a\) are often used to represent numbers, which allows us to form equations and solve for these unknowns.
One of the main goals in algebra is to determine the value of these unknown variables. For example, when you're asked to find \(f(a)\) in the problem we've reviewed, you need to evaluate the algebraic expression by substituting a given value (here, \(a\)) into the equation. This involves replacing variables consistently and carefully to arrive at a final expression or numerical outcome.
Understanding how to manipulate these expressions is crucial, as algebra forms the basis for more advanced topics in mathematics, including calculus, statistics, and more applied math areas.

Quadratic Functions

Quadratic functions are polynomial functions of degree two. They are typically in the form of \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The characteristic U-shaped curve that graphs a quadratic function is known as a parabola.
In our exercise, the function we evaluated, \(f(x) = 3x^2 - 2\), is a quadratic function. Here, \(a = 3\), \(b = 0\), and \(c = -2\). This function has the highest power of \(x\) as two, indicating it is quadratic.
Quadratic functions have several interesting properties:

• They have a vertex, which is the highest or lowest point of the parabola, depending on the sign of \(a\).
• They can open upwards or downwards, which is determined by whether \(a\) is positive or negative.
• Solving quadratic functions usually involves different methods like factoring, completing the square, or using the quadratic formula.

Substitution Method

The substitution method is helpful not only in evaluating functions but also in solving systems of equations. It involves solving one equation for one variable then substituting that expression into another equation.
In the context of our exercise, substitution was used to evaluate the function \(f(x) = 3x^2 - 2\) at \(x = a\). This process involves replacing \(x\) with \(a\), giving us the new expression \(f(a) = 3a^2 - 2\). Here's how it typically works:

• Identify the expression or equation where substitution is needed.
• Choose the value or expression to substitute for the given variable.
• Replace the variable in the equation with the substitution value or expression.
• Simplify, if necessary, to find a final solution.

Substitution is a simple yet powerful tool in algebra and helps us solve equations efficiently by isolating variables, making complex problems more manageable.