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Chapter 0: Problem 50

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

Short Answer

Expert verified
The value of \( g(-2) \) is \( -10 \).

Step by step solution

01

Identify the given function

The given function is \( g(x) = 4x - 2 \).
02

Substitute the value

To find \( g(-2) \), substitute \( x = -2 \) into the function \( g(x) \): \[ g(-2) = 4(-2) - 2 \]
03

Simplify the expression

Calculate the result of the substitution: \[ 4(-2) = -8 \] \[ -8 - 2 = -10 \]
04

Write the final answer

Thus, \( g(-2) = -10 \).

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Key Concepts

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

Substitution in Functions
When evaluating functions, substitution is a crucial step. It's the process of replacing the variable with a specific value. For example, if we have a function defined as \(\text{g(x) = 4x - 2}\), and we want to find \(\text{g(-2)}\), we simply replace every \(\text{x}\) in the function with \(\text{-2}\).

Here鈥檚 a step-by-step guide to substitution:
  • Identify the function you are working with
  • Determine the value you need to substitute
  • Replace the variable with this value
  • Simplify the resulting expression

In our example, we start with \(\text{g(x) = 4x - 2}\) and substitute \(\text{x = -2}\) into the function, resulting in \(\text{g(-2) = 4(-2) - 2}\).
Simplifying Expressions
Simplifying expressions involves performing all possible arithmetic operations to reduce an expression to its simplest form. Let's illustrate this with the function \(\text{g(x) = 4x - 2}\).

After substituting \(\text{x = -2}\) into \(\text{g(x)}\), we get \(\text{g(-2) = 4(-2) - 2}\). Now, follow these steps to simplify it:
  • First, handle the multiplication: \(\text{4(-2) = -8}\).
  • Next, perform the subtraction: \(\text{-8 - 2 = -10}\).

Through these steps, we simplify \(\text{g(-2) = -10}\). Remember, simplify in stages, tackling one operation at a time.
Absolute Value Functions
The absolute value function, represented by \(\text{|x|}\), measures the distance of a number from zero on a number line, ignoring the sign. An absolute value function is special because it turns negative numbers into positive ones.

If we have \(\text{k(x) = |x+3|}\), let's see how it responds to different values of \(\text{x}\):
  • For \(\text{x = -5}\), \(\text{k(-5) = |-5 + 3| = |-2| = 2}\).
  • For \(\text{x = 1}\), \(\text{k(1) = |1 + 3| = |4| = 4}\).
  • For \(\text{x = -3}\), \(\text{k(-3) = |-3 + 3| = |0| = 0}\).

The absolute value function always yields a non-negative result, reflecting its core property of measuring distance.

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Most popular questions from this chapter

A right triangle has vertices \((0,0), (1,0)\) , and \((1, \sqrt{3})\). Find the midpoint of the hypotenuse. Find the distance from the midpoint of the hypotenuse to each vertex.

Solve each problem. Perpendicular Tangents A circle of radius 5 intersects a circle of radius 12 so that the tangent lines at the points of intersection are perpendicular. What is the distance between the centers of the circles?

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

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=2 x-1 $$

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=-x+3 $$
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