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Chapter 0: Problem 51

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ k(5) $$

Short Answer

Expert verified
The value of k(5) is 8.

Step by step solution

01

- Understand the function k(x)

The function given is k(x) = |x + 3|. This is an absolute value function, which means it will always return a non-negative value.
02

- Substitute the value into the function

Substitute the given value, x = 5, into the function k(x). This means we will calculate k(5).
03

- Calculate the expression inside the absolute value

Inside the function, replace x with 5: |5 + 3|.
04

- Simplify the expression

Add the numbers inside the absolute value: 5 + 3 = 8.
05

- Apply the absolute value

Take the absolute value of 8, which is simply 8 because it is already a positive number: |8| = 8.

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Key Concepts

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

Understanding Absolute Value Function
The absolute value function is crucial in many mathematical problems. An absolute value of a number is its distance from zero on the number line. This is always a non-negative number. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5. The notation for the absolute value of x is |x|.
The absolute value function ensures that regardless of the input, the output is always positive unless the input is zero. This makes it useful in various scenarios where negative results are not meaningful or wanted.
Evaluating Functions
Evaluating functions is a fundamental concept in mathematics. It involves replacing the variable in the function with a given number and computing the result. Let's break it down using our original exercise:
  • We have the function \(k(x) = |x + 3|\).
To evaluate this function at \(x = 5\), we substitute 5 in place of x and solve the resulting expression:
  • First, replace x with 5: \(k(5) = |5 + 3|\).
  • Next, simplify the expression inside the absolute value: \(5 + 3 = 8\).
  • Finally, take the absolute value of 8, which is 8: \(|8| = 8\).
Thus, we find that \(k(5) = 8\). This step-by-step process of substitution and simplification is essential when working with functions.
Function Substitution
Function substitution is when you replace the variable in a function with a specific value or another expression. This is often used to evaluate functions at specific points or to combine different functions. In our given exercise, we substituted the number 5 into the function \(k(x)\). Here鈥檚 another quick example:
  • If you have \(f(x) = 3x^2 - x\) and you need \(f(2)\), you replace x with 2: \(f(2) = 3(2)^2 - 2 = 3 \times 4 - 2 = 12 - 2 = 10\).
Function substitution helps in finding specific values of complex functions easily. It's an essential skill for solving more advanced problems in mathematics and various applied fields.

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Most popular questions from this chapter

For each pair of variables determine whether \(a\) is a function of \(b\), \(b\) is a function of \(a\), or neither. \(a\) is the cost of mailing any first-class letter and \(b\) is its weight.

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=x^{2}-20 $$

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Capsize Control The capsize screening value \(C\) is an indicator of a sailboat's suitability for extended offshore sailing. \(C\) is determined by the function \(C=4 D^{-1 / 3} B,\) where \(D\) is the displacement of the boat in pounds and \(B\) is its beam (or width) in feet. Sketch the graph of this function for \(B\) ranging from 0 to \(20 \mathrm{ft}\), assuming that \(D\) is fixed at \(22,800 \mathrm{lb}\). Find \(C\) for the Island Packet \(40,\) which has a displacement of \(22,800 \mathrm{lb}\) and a beam of \(12 \mathrm{~ft}\) \(11 \mathrm{~in} .\) (Island Packet Yachts, www.ipy.com).

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