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Chapter 3: Problem 18

Evaluate the function at the indicated values. $$f(x)=x^{3}+2 x, \quad f(-2), f(1), f(0), f\left(\frac{1}{3}\right), f(0.2)$$

Short Answer

Expert verified
f(-2)=-12, f(1)=3, f(0)=0, f\left(\frac{1}{3}\right)=\frac{19}{27}, f(0.2)=0.408.

Step by step solution

01

Evaluate f(-2)

To find \( f(-2) \), substitute \( x = -2 \) into the function. Calculate: \[ f(-2) = (-2)^3 + 2(-2) = -8 - 4 = -12. \] Thus, \( f(-2) = -12 \).
02

Evaluate f(1)

Substitute \( x = 1 \) into the function. Calculate: \[ f(1) = 1^3 + 2(1) = 1 + 2 = 3. \] Thus, \( f(1) = 3 \).
03

Evaluate f(0)

Substitute \( x = 0 \) into the function. Calculate: \[ f(0) = 0^3 + 2(0) = 0. \] Thus, \( f(0) = 0 \).
04

Evaluate f(\(\frac{1}{3}\))

Substitute \( x = \frac{1}{3} \) into the function. Calculate: \[ f\left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^3 + 2\left(\frac{1}{3}\right) = \frac{1}{27} + \frac{2}{3}. \] To add these, convert \( \frac{2}{3} \) to \( \frac{18}{27} \) and add: \[ \frac{1}{27} + \frac{18}{27} = \frac{19}{27}. \] Thus, \( f\left(\frac{1}{3}\right) = \frac{19}{27} \).
05

Evaluate f(0.2)

Substitute \( x = 0.2 \) into the function. Calculate: \[ f(0.2) = (0.2)^3 + 2(0.2) = 0.008 + 0.4 = 0.408. \] Thus, \( f(0.2) = 0.408 \).

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Key Concepts

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

Polynomial Function
A polynomial function is a mathematical expression that involves the sum of powers of a variable. In simpler terms, it's an equation that consists of variables raised to whole number exponents. These exponents determine the degree of the polynomial.

Consider the function given, \( f(x) = x^3 + 2x \):
  • Term 1: \( x^3 \) is a cubic term; it determines the highest degree, making this a third-degree polynomial.
  • Term 2: \( 2x \) is a linear term. It adds a linear component to the polynomial.
Polynomial functions can describe a wide variety of shapes when graphed and are integral in modeling diverse real-world situations. Understanding the structure of polynomials helps in manipulating them, making calculations, and solving related problems.
Substitution Method
The substitution method is a technique used to evaluate mathematical expressions by replacing a variable with a given number. This method is simple but extremely powerful in determining specific outputs of a function based on inputs.

For example, if we have \( f(x) = x^3 + 2x \) and are asked to find \( f(-2) \), follow these steps:
  • Step 1: Replace \( x \) with \( -2 \) in the function.
  • Step 2: Perform the arithmetic operations: \( (-2)^3 + 2(-2) \).
  • Step 3: Simplify to find the result \( -12 \).
This method is not limited to numbers but can also be expanded to include expressions or other variables, enhancing its versatility.
Arithmetic Operations
Arithmetic operations are fundamental mathematical processes that help us manipulate numbers. They include addition, subtraction, multiplication, and division.

For the function \( f(x) = x^3 + 2x \), let's see how these operations play a role:
  • Multiplication: \( x^3 \) and \( 2x \) represent multiplication of the variable by itself or a constant.
  • Addition: Adding the results of these terms gives the final output of the function.
Understanding and applying these operations correctly ensures accurate evaluation of expressions. It's crucial to follow the order of operations or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to prevent errors.
Evaluating Expressions
Evaluating expressions refers to the process of determining the value of a function or expression given specific inputs. This task is essential in many fields like engineering, physics, and economics, where equations model real-world phenomena.

To evaluate the function \( f(x) = x^3 + 2x \) at different values:
  • Substitute the given value into the expression.
  • Perform the necessary operations (like in previous steps where we substituted and simplified terms).
  • The resulting number is the evaluation of the function at that point.
Knowing how to evaluate expressions accurately can provide insights into how variables affect outcomes, making it a vital skill in mathematical problem-solving.

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Most popular questions from this chapter

Express the function in the form \(f \circ g \circ h\) $$ G(x)=\frac{2}{(3+\sqrt{x})^{2}} $$

Revenue, cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per bumper sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that revenue \(=\) price per tiem \(\times\) number of items sold to express \(R(x),\) the revenue from an order of \(x\) stickers, as a product of two functions of \(x\)

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. $$ f(x)=x \cdot|x| $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{x}{x+1}, \quad g(x)=\frac{1}{x} $$

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x .\) $$ g(x)=\sqrt{x+3} $$
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