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Chapter 0: Problem 48

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(-4) $$

Short Answer

Expert verified
f(-4) = 52

Step by step solution

01

Understand the function f(x)

The function given is \( f(x) = 3x^2 - x \). This function needs to be evaluated at \( x = -4 \).
02

Substitute -4 into f(x)

Replace \( x \) with \(-4\) in the function \( f(x) \): \[ f(-4) = 3(-4)^2 - (-4) \]
03

Calculate \(-4\) squared

Calculate \( (-4)^2 \): \[ (-4)^2 = 16 \]
04

Multiply by 3

Multiply 16 by 3: \[ 3 \times 16 = 48 \]
05

Add the last term

Add 4 to 48: \[ 48 + 4 = 52 \]
06

Combine the results

So, the value of \( f(-4) \) is \[ 52 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, the function given in the exercise is a polynomial: \( f(x) = 3x^2 - x \). Polynomial functions are common in algebra and have different types based on their degrees. The degree of a polynomial is the highest power of the variable. In our case, the highest power of \( x \) is 2, so it is a second-degree polynomial, also known as a quadratic polynomial.
Function Evaluation
Function evaluation is the process of finding the output of a function for a specific input. To evaluate a function like \( f(x) = 3x^2 - x \) at a particular value of \( x \), you substitute the value into the function. For instance, when asked to find \( f(-4) \), you replace \( x \) in the equation with \( -4 \). This helps you understand how the function behaves at different points.
Substitution in Functions
Substitution in functions involves replacing the variable in the function with a given number. This step-by-step replacement allows you to compute the value of the function at that point. Consider the exercise where we have to find \( f(-4) \). We substitute \( -4 \) into the function:
  • Replace \( x \) with \( -4 \) in \( f(x) \)
  • Calculate \( f(-4) = 3(-4)^2 - (-4) \)
  • Simplify to get the value.
This substitution is straightforward but crucial for solving many algebraic problems.
Basic Algebraic Operations
In solving polynomial functions, you'll often use basic algebraic operations such as addition, subtraction, multiplication, and exponentiation. For our example, these steps break down as follows:
  • Square the input value: Calculate \( (-4)^2 \)
  • Multiply by the coefficient: \( 3 \times 16 \)
  • Combine like terms: Add \( 48 + 4 \)
  • Result: \( f(-4) = 52 \)
Understanding these operations helps simplify and solve polynomial equations efficiently.

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Most popular questions from this chapter

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