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Chapter 7: Problem 5

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ g(-2) $$

Short Answer

Expert verified
The value of \(g(-2)\) is \(-11\).

Step by step solution

01

Identify the function g(x)

The function provided is \(g(x) = -x^2 + 4x + 1\).
02

Substitute x with -2 in g(x)

Replace \(x\) with \(-2\) in the function: \(-(-2)^2 + 4(-2) + 1\).
03

Calculate \(-(-2)^2\)

Square \(-2\) to get 4, then multiply by -1 to get \(-4\).
04

Calculate \(4(-2)\)

Multiply 4 by \(-2\) to get \(-8\).
05

Add the results

Add \(-4\), \(-8\), and 1 to get \(-4 - 8 + 1 = -11\).
06

Final answer

The final value of \(g(-2)\) is \(-11\).

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Key Concepts

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

Function Substitution
Function substitution is an important concept in algebra. It involves replacing the variable in a function with a given number or another function. In the original exercise, we replaced the variable \( x \) in the function \( g(x) \) with \( -2 \). This process helps us to evaluate the function at specific points, which is useful for understanding the behavior of the function.
For instance:
If \( g(x) = -x^2 + 4x + 1 \),
and we need to find \( g(-2) \), we substitute \( x \) with \( -2 \):
\( g(-2) = -(-2)^2 + 4(-2) + 1 \)
This principle is very practical in math, physics, and engineering, where we often need to evaluate how changing one variable affects the entire system.
Polynomial Functions
Polynomial functions are algebraic expressions that involve terms in the form \( ax^n \), where \( a \) is a coefficient, \( x \) is the variable, and \( n \) is a non-negative integer exponent. The function \( g(x) = -x^2 + 4x + 1 \) is a polynomial function of degree 2 (also known as a quadratic function), because the highest exponent of \( x \) is 2.

The general form of a quadratic function is:
\[ ax^2 + bx + c \]
where \( a \), \( b \), and \( c \) are constants. In our example:
  • \( a = -1 \)
  • \( b = 4 \)
  • \( c = 1 \)
This function will produce a parabola when graphed. Understanding polynomial functions helps with interpreting graphs and solving equations efficiently.
Basic Algebraic Operations
Basic algebraic operations are the building blocks for solving more complex mathematical expressions. They include addition, subtraction, multiplication, and division. In the provided solution, we used several basic algebraic operations to simplify \( g(-2) \):

1. **Square the variable**: Squaring \( -2 \) to get \( 4 \).
\( (-2)^2 = 4 \)

2. **Multiply by coefficients**: Apply the coefficients from the polynomial function.
\( -1 \times 4 = -4 \)
\( 4 \times -2 = -8 \)

3. **Add the constants**: Finally, add all the results together.
\( -4 + (-8) + 1 = -11 \)
These operations are essential for evaluating and simplifying polynomial functions, and they are core skills in algebra.

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Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ f(4)-g(4) $$
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