Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=-x^{2}-x-6 ; x=3$$

Step-by-Step Solution

Answer

When \( x = 3 \), \( f(x) = -18 \).

Step 1: Understand the Function

The function given is \( f(x) = -x^2 - x - 6 \). It is a quadratic function in terms of \( x \). To find the value of \( f(x) \) at \( x = 3 \), we need to substitute 3 for \( x \) in the function.

Step 2: Substitute x with 3

Substitute \( x = 3 \) into the function: \[ f(3) = -(3)^2 - 3 - 6 \].

Step 3: Calculate Powers

Calculate \( 3^2 \): \[ 3^2 = 9 \].

Step 4: Apply Negative Sign

Apply the negative sign to the squared term: \[ -(3)^2 = -9 \].

Step 5: Evaluate the Expression

Now evaluate the expression with the calculated powers: \[ f(3) = -9 - 3 - 6 \].

Step 6: Simplify the Expression

Simplify the expression step-by-step: \(-9 - 3 = -12 \)\(-12 - 6 = -18 \)Thus, \( f(3) = -18 \).

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra and can be represented by the general form:

1. Standard Form: \( f(x) = ax^2 + bx + c \)

Here, \( a \), \( b \), and \( c \) are constants, and the highest degree of \( x \) is 2. This form is called "quadratic" because it involves a square term, \( x^2 \).
Quadratic functions often model real-world situations such as projectile motion, population growth, and material stress. What's distinctive about these functions is their characteristic U-shaped graph called a parabola, which can open upwards or downwards depending on the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

For the given exercise, the function \( f(x) = -x^2 - x - 6 \) is a quadratic function with a negative leading coefficient, indicating that its parabola opens downwards. Understanding these characteristics helps us grasp how the outputs of the function change as the inputs, \( x \), vary.

Substitution Method

The substitution method is a straightforward technique used in mathematics to simplify functions and equations by replacing variables with known values. It's particularly helpful for evaluating functions at specific points.
The basic idea is simple:

• Identify the variable you want to substitute.
• Replace the variable with its given value in the equation or function.
• Perform the necessary calculations to find the result.

In our exercise, we substitute \( x = 3 \) into the quadratic function \( f(x) = -x^2 - x - 6 \). This involves replacing every occurrence of \( x \) in the function with 3, transforming it into \( f(3) = -(3)^2 - 3 - 6 \).
By using substitution, we transform the function into an arithmetic problem. This allows us to evaluate the function at that specific input value effectively.

Expression Simplification

Expression simplification is an essential skill in algebra that makes problems easier to solve by reducing complex mathematical expressions to simpler forms. It involves performing operations such as:

• Calculating powers and roots
• Applying arithmetic operations
• Combining like terms

For our given function, after substituting \( x = 3 \), we simplify the expression \( f(3) = -(3)^2 - 3 - 6 \).
Here are the steps in simplification:

• Calculate powers: \( 3^2 = 9 \)
• Apply the negative sign: \( -(3)^2 = -9 \)
• Combine the results: Proceed with the arithmetic \( -9 - 3 = -12 \)
• Then simplify further: \( -12 - 6 = -18 \)

The result, \( f(3) = -18 \), is now a single number, much simpler than the initial quadratic expression. Mastering expression simplification can make a lot of algebraic and calculus problems more manageable and set the foundation for solving more complex mathematical equations.