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Chapter 0: Problem 49

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

Short Answer

Expert verified
g(-1) = -6

Step by step solution

01

Identify the function to use

The problem requires you to find the value of function g(x) at x = -1. Use the given function g(x) = 4x - 2.
02

Substitute x with -1

Replace the variable x in the function g(x) with -1. So, g(-1) = 4(-1) - 2.
03

Simplify the expression

Calculate the value inside the function: g(-1) = 4(-1) - 2 = -4 - 2 = -6.

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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 a process where you replace the variable in a function with a specific value. For example, if you have a function defined as g(x) = 4x - 2, to find g(-1), you substitute -1 for every x in the expression. This is completed step-by-step in the following way:
  • Identify the function to use.
  • Substitute x with the given value (-1).
  • Simplify the equation.
In this exercise, we need to find g(-1). Since g(x) = 4x - 2, we replace x with -1 and follow the steps to simplify: g(-1) = 4(-1) - 2 = -4 - 2 = -6. This gives us the value -6 for g(-1).
Linear Functions
A linear function is any function that can be written in the form f(x) = mx + b, where m and b are constants. Linear functions graph as straight lines and have a constant slope m. In the function g(x) = 4x - 2, the coefficient of x is 4, which means the slope of the line is 4, and the constant term is -2, which represents the y-intercept.
Understanding the components of a linear function is crucial:
  • The slope (m) shows how steep the line rises or falls.
  • The y-intercept (b) indicates where the line crosses the y-axis.
With these insights, you can quickly evaluate linear functions for any given x value, as done in the original exercise with g(-1).
Absolute Value Function
An absolute value function is defined as f(x) = |x|, which represents the distance of x from zero on the number line without considering the direction. This function always results in a non-negative value.
Consider the function k(x) = |x + 3|. Here, the expression inside the absolute value is x + 3. Regardless of whether x is positive or negative, the |x + 3| will always return a non-negative result.
  • If x + 3 is positive or zero, |x + 3| = x + 3.
  • If x + 3 is negative, |x + 3| = -(x + 3).
By understanding how the absolute value function works, you can evaluate these functions accurately for any given x value.

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