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Chapter 1: Problem 30

Evaluate (if possible) the function at each specified value of the independent variable and simplify. \(f(x)=|x|+4\) (a) \(f(2)\) (b) \(f(-2)\) (c) \(f\left(x^{2}\right)\)

Short Answer

Expert verified
The evaluated function outputs are: \n a) \(f(2)= 6\), \n b) \(f(-2)= 6\), \n c) \(f\left(x^{2}\right)= x^{2} + 4\).

Step by step solution

01

Evaluating For Positive x

When a positive number is inserted for x. This means, for \(f(2)\), the value in the absolute value function would remain positive. Hence, plug in 2 into the function to get \(f(2)= |2| + 4 = 2 + 4= 6\).
02

Evaluating For Negative x

When a negative number is inserted for x, it changes. This is because the absolute value converts negative numbers to their corresponding positive values. For \(f(-2)\), the absolute value would change -2 to 2. Hence, \(f(-2) = |-2| + 4 = 2 + 4 = 6\).
03

Evaluating For An Algebraic Expression

In the case of an algebraic expression such as \(f\left(x^{2}\right)\), substitute \(x^{2}\) for x in the given function, resulting in \(f\left(x^{2}\right) = \left|x^{2}\right| + 4. Since the square of any real number is positive, this simplifies to \(f\left(x^{2}\right) = x^{2} + 4\).

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Key Concepts

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

Independent Variable
When dealing with mathematical functions, it's essential to understand the role of the independent variable. This is a variable that stands alone and isn't affected by other variables. In the function \(f(x) = |x| + 4\), the independent variable is \(x\). It is called 'independent' because you can choose any value for \(x\), and the function will provide a result based on that choice.
  • When \(x = 2\), the function becomes \(f(2)\).
  • When \(x = -2\), the function becomes \(f(-2)\).
  • When \(x = x^2\) for symbolic evaluation, the result changes to \(f(x^2)\).
Understanding this concept helps grasp the flexibility and application of functions in algebra.
Function Evaluation
Function evaluation is the process of finding the value of a function given a specific input. This involves substituting the given value of the independent variable into the function's equation. As demonstrated in the exercise, function evaluation follows these steps:
1. **Substitute** the value into the equation. For our function \(f(x) = |x| + 4\), substitute the value of \(x\) with the number given.
  • For \(f(2)\), replace \(x\) with 2 to get \(f(2) = |2| + 4\).
  • For \(f(-2)\), replace \(x\) with -2 leading to \(f(-2) = |-2| + 4\).
2. **Solve** the equation. This involves evaluating the absolute value and performing any arithmetic operations.
Function evaluation is crucial for verifying function behavior at specific points.
Algebraic Expression
Algebraic expressions involve numbers, variables, and operations. They represent a set of values dependent on the variables' inputs. Sometimes, instead of substituting numbers into a function, you substitute an algebraic expression. This was the case with \(f(x^2)\) in our example.
  • Instead of setting \(x = 2\) or \(x = -2\), we substitute \(x^2\) in place of \(x\).
  • The function becomes \(f(x^2) = |x^2| + 4\) which simplifies to \(x^2 + 4\) because \(x^2\) is always non-negative.
Understanding algebraic expressions and their manipulation is foundational in algebra, allowing for the evaluation and simplification of functions beyond simple numerical substitutions.

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Most popular questions from this chapter

Assume that \(y\) is directly proportional to \(x .\) Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x .\) $$x=6, y=580$$

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(v\) varies jointly as \(p\) and \(q\) and inversely as the square of s. \((v=1.5 \text { when } p=4.1, q=6.3, \text { and } s=1.2 .)\)

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The annual gross ticket sales \(S\) (in millions of dollars) for Broadway shows in New York City from 1995 through 2011 are given by the following ordered pairs. $$\begin{aligned} &(1995,406) \quad(2004,771)\\\ &(1996,436) \quad(2005,769)\\\ &(1997,499) \quad(2006,862)\\\ &(1998,558) \quad(2007,939)\\\ &(1999,588) \quad(2008,938)\\\ &(2000,603) \quad(2009,943)\\\ &(2001,666) \quad(2010,1020)\\\ &(2002,643) \quad(2011,1080)\\\ &(2003,721) \end{aligned}$$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t=5\) represent 1995 (b) Use the regression feature of the graphing utility to find the equation of the least squares regression line that fits the data. (c) Use the graphing utility to graph the scatter plot you created in part (a) and the model you found in part (b) in the same viewing window. How closely does the model represent the data? (d) Use the model to predict the annual gross ticket sales in 2017 (e) Interpret the meaning of the slope of the linear model in the context of the problem.
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