Problem 36 - Algebra 2

Question

Find each value if \(f(x)=3 x-5\) and \(g(x)=x^{2}-x\) \(g(3)\)

Step-by-Step Solution

Answer

g(3) = 6

Step 1: Understand the Function g(x)

We are given the function \( g(x) = x^2 - x \). This means for any value of \( x \), \( g(x) \) is calculated by squaring \( x \) and then subtracting \( x \) from the result.

Step 2: Substitute x with 3

To find \( g(3) \), substitute \( x = 3 \) into the function \( g(x) \): \( g(3) = 3^2 - 3 \).

Step 3: Calculate 3 Squared

First, find the square of 3. \( 3^2 = 9 \).

Step 4: Subtract 3

Now subtract \( 3 \) from \( 9 \). This gives \( 9 - 3 = 6 \).

Step 5: Conclude the Calculation

Thus, the value of \( g(3) \) is 6.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves finding the value of a function at a particular input. When we talk about a function like \(g(x) = x^2 - x\), it's like having a machine where you can input a value for \(x\), and it will output a result. For example, if you input \(x = 3\) into this function, the machine will perform its operations and give you the result.To evaluate a function:

• Identify the function you are dealing with (e.g., \(g(x) = x^2 - x\)).
• Substitute the given value into the function (e.g., find \(g(3)\)).
• Perform any operations needed to reach the final output.

Understanding function evaluation helps simplify problems and is often the first step in solving more complex algebraic equations.

Quadratic Functions

Quadratic functions are a special category of polynomials, characterized by the fact that their highest power of the variable is 2. The general form of a quadratic function is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. They are called "quadratic" because of the term \(x^2\).Quadratic functions have a distinctive U-shaped graph known as a parabola:

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.
• The vertex of the parabola is its minimum or maximum point, depending on the direction it opens.

In the context of the problem \(g(x) = x^2 - x\), we see a simple quadratic where \(a = 1\), \(b = -1\), and \(c = 0\). These simple forms make substitution easier, as we can quickly compute values.

Substitution Method

The substitution method in mathematics is a powerful technique used to evaluate functions or solve equations. It involves replacing variables with specific numbers or expressions, simplifying the total problem.Here's how to use substitution:

• Identify the variable in the function you want to replace (e.g., \(x\) in \(g(x) = x^2 - x\)).
• Insert the given number in place of the variable (e.g., substitute \(x = 3\) to find \(g(3)\)).
• Perform the necessary calculations by following the order of operations: powers first, followed by multiplication, division, addition, and subtraction.

Using substitution simplifies the process, helping break down complicated functions step by step. It is particularly useful in algebra when finding specific values or solving systems of equations.