Problem 37 - College Algebra

Question

Given the function \(f(x)=x^{2}-3 x\) a. Evaluate \(f(5)\). b. Solve \(f(x)=4\)

Step-by-Step Solution

Answer

a. \(f(5) = 10\) b. \(x = 4\) or \(x = -1\)

Step 1: Evaluate the Function at a Given Point

To find \( f(5) \), substitute \( x = 5 \) into the function \( f(x) = x^2 - 3x \). Thus, \( f(5) = (5)^2 - 3(5) = 25 - 15 = 10 \).

Step 2: Set Function Equal to 4

To find \( x \) when \( f(x) = 4 \), set the function equal to 4: \[ x^2 - 3x = 4 \]

Step 3: Rearrange the Equation

Rearrange the equation to form a standard quadratic equation: \[ x^2 - 3x - 4 = 0 \]

Step 4: Factor the Quadratic Equation

Factor the quadratic equation \( x^2 - 3x - 4 = 0 \). It can be factored as: \[ (x - 4)(x + 1) = 0 \]

Step 5: Solve for x

Set each factor equal to zero and solve for \( x \):1. \( x - 4 = 0 \) gives \( x = 4 \).2. \( x + 1 = 0 \) gives \( x = -1 \).

Step 6: Verify the Solutions

Verify the solutions by substituting them back into \( f(x) \):\( f(4) = 4^2 - 3(4) = 16 - 12 = 4 \), correct.\( f(-1) = (-1)^2 - 3(-1) = 1 + 3 = 4 \), correct.

Key Concepts

Function Evaluation

Evaluating a function at a specific point means finding the output value when we substitute an input value into the function. Here, the function is given as \( f(x) = x^2 - 3x \).

To evaluate \( f(5) \), we substitute \( x = 5 \) into the function's equation.

• Substitute: \( f(5) = (5)^2 - 3(5) \)
• Calculate: \( 25 - 15 = 10 \)

Thus, the value of \( f(5) \) is 10.
Remember, function evaluation is simply replacing the variable with a given number and performing basic arithmetic operations.

Solving Quadratic Equations

Solving a quadratic equation involves finding the values of the variable that make the equation true. A quadratic equation has the form \( ax^2 + bx + c = 0 \). In our exercise, we need to solve the equation derived from the function \( x^2 - 3x = 4 \).

To solve it, first, rearrange it into the standard quadratic form:

• Subtract 4 from both sides: \( x^2 - 3x - 4 = 0 \)

Once we have the equation in standard form, we can begin solving it. This usually involves methods like factoring, the quadratic formula, or completing the square.

Factoring Quadratic Equations

Factoring is a method for solving quadratics by expressing the equation as a product of two binomials. For the equation \( x^2 - 3x - 4 = 0 \), the goal is to find two numbers that multiply to \(-4\) and add to \(-3\).

The potential factors are determined to be \( -4 \) and \( +1 \). Thus, we can write the quadratic as:

• Factor: \( (x - 4)(x + 1) = 0 \)

Factoring allows us to break down the quadratic into more manageable linear pieces, which can then be solved separately by setting each factor equal to zero.

Verifying Solutions

After finding potential solutions to a quadratic equation, it's crucial to verify them to ensure correctness. Once we have factored the equation into \( (x - 4)(x + 1) = 0 \), we solve to find:

• \( x - 4 = 0 \) gives \( x = 4 \)
• \( x + 1 = 0 \) gives \( x = -1 \)

Verification involves substituting these \( x \)-values back into the original function \( f(x) = x^2 - 3x \):

• For \( x = 4 \): \( f(4) = 4^2 - 3(4) = 16 - 12 = 4 \)
• For \( x = -1 \): \( f(-1) = (-1)^2 - 3(-1) = 1 + 3 = 4 \)

Since both substitutions result in 4, we have correctly verified our solutions. This step checks our work and confirms the accuracy of the solutions.