Problem 94 - Algebra 2

Question

PREREQUISITE SKILL. Evaluate each function for the given value. $$ f(x)=x^{2}+2 x-3, x=2 $$

Step-by-Step Solution

Answer

The value of the function \( f(x) \) at \( x = 2 \) is 5.

Step 1: Understand the Function

The function given is \( f(x) = x^2 + 2x - 3 \). This is a quadratic function, which means that it includes a term with \( x^2 \). The goal is to evaluate this function for a specific value of \( x \).

Step 2: Substitute the Value of x

Substitute \( x = 2 \) into the function \( f(x) = x^2 + 2x - 3 \). This means replacing every \( x \) in the function with the number 2: \[ f(2) = (2)^2 + 2(2) - 3 \].

Step 3: Calculate the Square Term

First, calculate \( (2)^2 \), which is 4. So the expression is now: \[ f(2) = 4 + 2(2) - 3 \].

Step 4: Multiply in the Linear Term

Next, calculate the multiplication \( 2 \times 2 \), which gives 4. Now the expression becomes: \[ f(2) = 4 + 4 - 3 \].

Step 5: Perform the Arithmetic

Now, perform the addition and subtraction from left to right: First, add 4 + 4 to get 8. Then subtract 3 from 8 to get 5. Thus, \( f(2) = 5 \).

Key Concepts

Quadratic Functions

Quadratic functions form the backbone of many algebraic concepts. They are beautiful equations that express a curve known as a parabola when graphed on a coordinate plane. A quadratic function can be recognized by its highest term being something squared. This means the equation always looks like something involving \( x^2 \).
For example, our function \( f(x) = x^2 + 2x - 3 \) is a quadratic function. It contains the term \( x^2 \), making it quadratic.
Here are some characteristics of quadratic functions:

• The graph of a quadratic function is a parabola, which can open upwards or downwards.
• There is usually a single vertex, acting as the maximum or minimum point of the parabola.
• The function can cross the x-axis at zero, one, or two points which are the roots of the quadratic equation.

Quadratic functions are highly useful in various real-world applications such as physics, engineering, and economics.

Function Evaluation

Function evaluation is a fundamental concept in algebra, especially when working with different types of functions. It’s the process of calculating the output of a function for a particular input.
In simple terms, it means plugging in a value for \( x \) in the function and simplifying to find the answer. For our function \( f(x) = x^2 + 2x - 3 \), evaluating the function at \( x = 2 \) involves finding \( f(2) \).
When evaluating functions, follow these steps:

• Start with the given function.
• Substitute the given value into the function in place of the variable \( x \).
• Simplify the expression to find the result.

By understanding function evaluation, you can solve more complex problems and enhance your problem-solving skills in mathematics.

Substitution Method

The substitution method is a straightforward technique used in the evaluation of functions. It involves replacing variables in an equation with particular numbers. When working with functions, the substitution method helps us find specific outputs by replacing \( x \) with provided values.
For example, in evaluating \( f(x) = x^2 + 2x - 3 \) at \( x = 2 \), we substitute 2 every time we see \( x \). This is known as plugging the value into the function.
Here's a quick guide on how to apply the substitution method:

• Identify the function and the value to substitute.
• Insert the given number in place of each variable within the function.
• Perform the necessary arithmetic operations to conclude.

Remember, accurate substitution and calculation are crucial to obtaining the correct result. This method makes problem-solving more manageable, especially when dealing with polynomial and algebraic functions.