Problem 54 - Algebra 2

Question

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(2) $$

Step-by-Step Solution

Answer

The value of \(f(2)\) is \(0\).

Step 1: Understand the Function

The function given is \(f(x) = x^2 - 5x + 6\). This is a quadratic function, and we will use it to evaluate the value of \(f(x)\) at \(x = 2\).

Step 2: Substitute 2 into the Function

To find \(f(2)\), substitute \(2\) for every \(x\) in the function. So we have \(f(2) = (2)^2 - 5(2) + 6\).

Step 3: Simplify the Expression

Calculate the expression we formed: \((2)^2 - 5(2) + 6 = 4 - 10 + 6\).

Step 4: Perform the Arithmetic

Solve the arithmetic step-by-step: \(4 - 10 = -6\), then \(-6 + 6 = 0\).

Step 5: Write the Final Result

The value of \(f(2)\) is determined to be \(0\). Therefore, \(f(2) = 0\).

Key Concepts

Evaluating Functions

When we talk about evaluating functions, what we’re really doing is finding the output value of a function for a given input. For example, given the quadratic function \( f(x) = x^2 - 5x + 6 \), we want to find the value of \( f(2) \). This simply means we are substituting the input value of \( x = 2 \) into the function to see what it produces for this input.

• Think of the function as a machine where you input a value (in this case, \( x=2 \)) and it processes it using the function rule (\( x^2 - 5x + 6 \)) to produce an output.
• Evaluating functions involves different types of functions, such as linear, quadratic, or exponential, each with their own characteristic shape and formula.
• In our specific example of a quadratic function, the form is \( ax^2 + bx + c \) where \( a, b, \) and \( c \) are constants.

By evaluating the function, we retrieve a numerical result demonstrating how the function behaves with a specific input.

Substitution Method

The substitution method is a straightforward technique where we replace the variable in the function with a specific number. This is what we did in the problem with \( f(x) = x^2 - 5x + 6 \) when we wanted to find \( f(2) \).

• Substitute \( x = 2 \) directly into the function to get \( f(2) = (2)^2 - 5(2) + 6 \).
• Each instance of the variable \( x \) in the equation is replaced with \( 2 \).
• Once the value is substituted, all that's left is to follow through with calculation.

This simple replacement allows us to transform a variable expression into a basic numerical one, which sets the stage for the fun math part: solving the arithmetic operation.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics, involving addition, subtraction, multiplication, and division. After substituting a value into a function, you often perform arithmetic operations to simplify the expression and reach a final result.

• In the function \( f(2) = (2)^2 - 5(2) + 6 \), the arithmetic process involves multiple steps: calculate \( (2)^2 \), then \( -5 \times 2 \), followed by adding and subtracting the values.
• First, calculate \( (2)^2 = 4 \).
• Next, perform \( 5 \times 2 = 10 \) and make it negative: \( -10 \).
• Finally, perform \( 4 - 10 \) which equals \( -6 \), then \( -6 + 6 = 0 \).

Through these operations, we establish the value of \( f(2) \). Each step in the arithmetic operation plays a crucial role, showing how numbers interact in the context of a function.