Question

Find \(f(-3)\) for \(f(x)=-2 x^2-7 x+9\) A. -30 B. -18 C. 12 D. 48

Short answer

The answer is C. 12.

Step-by-step solution

- Identify the function and the input

The given function is \(f(x) = -2x^2 - 7x + 9\). The input value to be used in the function is \(-3\).

- Substitute the input into the function

Substitute \(-3\) for \(x\) in the function: \(f(-3) = -2(-3)^2 - 7(-3) + 9\).

- Calculate the square term

Calculate \((-3)^2\): \((-3)^2 = 9\). This gives \(f(-3) = -2(9) - 7(-3) + 9\).

- Multiply the coefficients

Multiply the coefficients in the expression: \(-2 \times 9 = -18\) and \(-7 \times (-3) = 21\). This gives \(f(-3) = -18 + 21 + 9\).

- Combine the terms

Add the terms together: \(-18 + 21 = 3\) and, \(3 + 9 = 12\). Therefore, \(f(-3) = 12\).

- Select the answer

From the options given, the correct answer is C. 12.

Key concepts

Substitution in Functions

To solve function evaluation problems, one of the key steps is **substitution**. In this step, we replace the variable in the function with a specific value. This allows us to determine the function's output for that input. For example, if we have a function defined by the equation \(f(x) = -2x^2 - 7x + 9\), and we want to find \(f(-3)\), we substitute \(-3\) into the function wherever \(x\) appears.

When substituting, it is essential to carefully follow the arithmetic rules to ensure accuracy. Always use parentheses to ensure the operations are carried out correctly, especially when dealing with negative numbers and powers.

By correctly substituting \(-3\) for \(x\) into \(f(x)\), we get:

• \(f(-3) = -2(-3)^2 - 7(-3) + 9\)

This substitution helps us proceed to the next steps where we simplify and solve for the final result.

Polynomial Functions

Polynomial functions like the one in this exercise are composed of terms that include constants and variables raised to non-negative integer powers. In the function \(f(x) = -2x^2 - 7x + 9\), each term represents a part of a polynomial.

Polynomials are often classified based on their degree, which is the highest power of the variable in the function. In our case, the highest power is 2 (from the term \(-2x^2\)), making this a quadratic polynomial.

Polynomial functions are important in various fields of mathematics and applied sciences because they are simple yet flexible models that can represent complex behaviors. Understanding polynomial functions requires knowing how to handle each term individually and then combining them to get the overall result.

When we evaluate a polynomial function at a specific point, like substituting \(-3\) for \(x\), each term involving \(x\) must be recalculated to reflect this new input, and then they are summed up to find the function's value.

Arithmetic Operations

Arithmetic operations involve basic mathematical functions such as addition, subtraction, multiplication, and division. When evaluating a function, correctly performing these operations is crucial. Let’s break down the steps used in the exercise:

• Substitution: We substitute \(-3\) in place of \(x\) in the function.

\(f(-3) = -2(-3)^2 - 7(-3) + 9\)

• Exponentiation: Calculate the square term. \((-3)^2 = 9\)

\(f(-3) = -2(9) - 7(-3) + 9\)

• Multiplication: Multiply the coefficients. \(-2 \times 9 = -18\) and \(-7 \times (-3) = 21\)

\(f(-3) = -18 + 21 + 9\)

• Addition/Subtraction: Combine the terms. \(-18 + 21 = 3\) and then \(3 + 9 = 12\)

This structured approach ensures the correct evaluation of the function. Each arithmetic operation contributes to the final result, and careful execution is necessary to avoid errors. By following these steps, you can confidently solve function evaluation problems.