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Chapter 0: Problem 58

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

Short Answer

Expert verified
g(b) = 4b - 2

Step by step solution

01

Identify the function

The function that needs to be evaluated is the function g(x). The definition given is: \[ g(x) = 4x - 2 \]
02

Substitute the value

Substitute the value of b into the function g(x). This means replacing x with b in the expression for g(x): \[ g(b) = 4b - 2 \]
03

Simplify the expression

The function g(b) is already in its simplest form once b is substituted. Thus, we have: \[ g(b) = 4b - 2 \]

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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 the process of replacing the variable in a function with a given value or another expression. It allows us to evaluate the function at specific points or transform it based on new variables.

To perform function substitution, follow these steps:
  • Identify the function: Determine which function you're working with. This means knowing the formula or rule that defines the function.
  • Substitute the value: Replace the variable in the function with the given value or expression.

In our exercise, we have the function \(\text{g(x) = 4x - 2}\). To find \(\text{g(b)}\), replace \(\text{x}\) with \(\text{b}\) to get \(\text{g(b) = 4b - 2}\). This simple step changes the function based on the new variable.
simplifying expressions
Simplifying expressions involves reducing them to their simplest form so they are easier to work with. In the context of functions, once you've performed the substitution, you may need to simplify what remains.

In our example, the function \(\text{g(x) = 4x - 2}\) is already quite straightforward. After substituting \(\text{x}\) with \(\text{b}\), the expression becomes \(\text{g(b) = 4b - 2}\). There are no additional steps needed to simplify because both terms are already in their basic form.

However, in more complex situations, you might need to:
  • Combine like terms
  • Factor expressions
  • Simplify fractions
  • Perform arithmetic operations
By breaking down the problem and simplifying each piece, you can make even the most complex functions easier to understand.
absolute value function
The absolute value function is a special type of function that measures the distance of a number from zero on the number line, regardless of direction. This means it always outputs a non-negative value.

For a function \(\text{k(x)}\), if the absolute value is applied, such as \(\text{k(x) = |x + 3|}\), the output is always positive, whether \(\text{x + 3}\) is positive or negative.

To evaluate an absolute value function, keep these points in mind:
  • Calculate the expression inside the absolute value symbol.
  • If the result is positive or zero, the absolute value is the result itself.
  • If the result is negative, change the sign to make it positive.

For example, if \(\text{x = -5}\), then \(\text{k(x) = |-5 + 3| = |-2| = 2}\). Regardless of the initial sign, the absolute value ensures the final result is positive.

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Most popular questions from this chapter

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=|x|-40 $$

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

Bore and Stroke The displacement \(D\) of an engine is given by the function $$ D=\frac{\pi}{4} B^{2} \cdot S \cdot N $$ where \(B\) is the bore (the diameter of a cylinder), \(S\) is the stroke (distance the piston moves), and \(N\) is the number of cylinders. The two- cylinder Harley-Davidson Evo engine has a bore of 3.498 in. and a stroke of 4.250 in. Find its displacement.

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=x^{2}+50 $$

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