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Chapter 0: Problem 20

A function \(f\) is given by $$ f(x)=3 x+2 $$ This function takes a number \(x\), multiplies it by 3 , and adds 2 a) Complete this table. $$ \begin{array}{|c|c|c|c|c|} \hline x & 4.1 & 4.01 & 4.001 & 4 \\ \hline f(x) & & & & \\ \hline \end{array} $$ b) Find \(f(5), f(-1), f(k), f(1+t),\) and \(f(x+h)\)

Short Answer

Expert verified
a) 14.3, 14.03, 14.003, 14; b) 17, -1, 3k+2, 5+3t, 3x+3h+2.

Step by step solution

01

Calculate f(4.1)

To find \( f(4.1) \), substitute \( x = 4.1 \) into the function: \[ f(4.1) = 3(4.1) + 2 \]Calculate the multiplication:\[ 3(4.1) = 12.3 \]Then add 2:\[ 12.3 + 2 = 14.3 \]Thus, \( f(4.1) = 14.3 \).
02

Calculate f(4.01)

Substitute \( x = 4.01 \) into the function: \[ f(4.01) = 3(4.01) + 2 \]Calculate the multiplication:\[ 3(4.01) = 12.03 \]Then add 2:\[ 12.03 + 2 = 14.03 \]Thus, \( f(4.01) = 14.03 \).
03

Calculate f(4.001)

Substitute \( x = 4.001 \) into the function: \[ f(4.001) = 3(4.001) + 2 \]Calculate the multiplication:\[ 3(4.001) = 12.003 \]Then add 2:\[ 12.003 + 2 = 14.003 \]Thus, \( f(4.001) = 14.003 \).
04

Calculate f(4)

Substitute \( x = 4 \) into the function: \[ f(4) = 3(4) + 2 \]Calculate the multiplication:\[ 3(4) = 12 \]Then add 2:\[ 12 + 2 = 14 \]Thus, \( f(4) = 14 \).
05

Find f(5)

Substitute \( x = 5 \) into the function: \[ f(5) = 3(5) + 2 \]Calculate the multiplication:\[ 3(5) = 15 \]Then add 2:\[ 15 + 2 = 17 \]Thus, \( f(5) = 17 \).
06

Find f(-1)

Substitute \( x = -1 \) into the function:\[ f(-1) = 3(-1) + 2 \]Calculate the multiplication:\[ 3(-1) = -3 \]Then add 2:\[ -3 + 2 = -1 \]Thus, \( f(-1) = -1 \).
07

Find f(k)

Substitute \( x = k \) into the function:\[ f(k) = 3k + 2 \]Thus, \( f(k) = 3k + 2 \).
08

Find f(1+t)

Substitute \( x = 1 + t \) into the function:\[ f(1+t) = 3(1 + t) + 2 \]Distribute the 3:\[ 3 \cdot 1 + 3 \cdot t = 3 + 3t \]Then add 2:\[ 3 + 3t + 2 = 5 + 3t \]Thus, \( f(1+t) = 5 + 3t \).
09

Find f(x+h)

Substitute \( x = x + h \) into the function:\[ f(x+h) = 3(x + h) + 2 \]Distribute the 3:\[ 3x + 3h \]Then add 2:\[ 3x + 3h + 2 \]Thus, \( f(x+h) = 3x + 3h + 2 \).

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Key Concepts

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

Function Evaluation
Evaluating a function is the process of finding the value of a function at a specific input value. For the function \( f(x) = 3x + 2 \), evaluating the function requires substituting the given input value for \( x \) and then performing the arithmetic operations that the function prescribes. Let's go over some key steps:
  • Substitute: Place the input value into the function in place of \( x \).
  • Multiply: Follow the order of operations by first multiplying the input value by the coefficient, which is \( 3 \) in this case.
  • Add: Once multiplication is complete, add \( 2 \) to the result from the multiplication step.
This step-by-step replacement and arithmetic operation allow you to determine the value of the function for any \( x \). It's like following a recipe: substitute, multiply, and add to get your result!
Table of Values
Creating a table of values helps visualize how change in input affects the output in a linear function. When you have a function like \( f(x) = 3x + 2 \), you can choose several \( x \) values to plug into the function, calculate \( f(x) \), and then record these results in a table.

This process can help spot patterns over a sequence of calculations:

  • Pick an input value \( x \).
  • Evaluate \( f(x) \) as described in the function evaluation section.
  • Write these input-output pairs in a table format to observe how \( f(x) \) behaves as \( x \) changes.
Through this, you can appreciate the linearity of a function like \( f(x) = 3x + 2 \) by observing how outputs increase or decrease proportionally with inputs.
Function Operations
Function operations involve expanding or simplifying expressions when the input is not just a simple number, but potentially another expression. For example, when evaluating \( f(1+t) \) for the function \( f(x) = 3x + 2 \), consider the following steps:
  • Substitute \( 1 + t \) into the function in place of \( x \).
  • Distribute the \( 3 \) across terms within the parentheses, resulting in \( 3(1 + t) = 3 + 3t \).
  • Add \( 2 \) to the expression, giving \( 5 + 3t \).
Similarly, for \( f(x + h) \), replace, distribute, and combine terms to obtain \( 3x + 3h + 2 \). These operations not only show how to evaluate functions with expressions but also highlight the importance of distributive property and algebraic manipulation.

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Most popular questions from this chapter

The annual interest rate \(r,\) when compounded more than once a year, results in a slightly higher yearly interest rate; this is called the annual (or effective) yield and denoted as Y. For example, \$1000 deposited at 5\%, compounded monthly for 1 yr \((12\) months \(),\) will have a value of \(A=1000\left(1+\frac{0.05}{12}\right)^{12}=\$ 1051.16 .\) The interest earned is \(\$ 51.16 / \$ 1000,\) or \(0.05116,\) which is \(5.116 \%\) of the original deposit. Thus, we say this account has a yield of \(Y=0.05116,\) or \(5.116 \% .\) The formula for annual yield depends on the annual interest rate \(r\) and the compounding frequency \(n:\) \(Y=\left(1+\frac{r}{n}\right)^{n}-1.\) For Exercises 41-48, find the annual yield as a percentage, to two decimal places, given the annual interest rate and the compounding frequency. Chris is considering two savings accounts: Sierra Savings offers \(5 \%,\) compounded annually, on savings accounts, while Foothill Bank offers \(4.88 \%,\) compounded weekly. a) Find the annual yield for both accounts. b) Which account has the higher annual yield?

The amount of money, \(A(t),\) in \(a\) savings account that pays 3\% interest, compounded quarterly for \(t\) years, with an initial investment of \(P\) dollars, is given by $$ A(t)=P\left(1+\frac{0.03}{4}\right)^{4 t} $$ If \(\$ 500\) is invested at \(3 \%\), compounded quarterly, how much will the investment be worth after 2 yr?

Rewrite each of the following as an equivalent expression using radical notation. $$ t^{-2 / 5} $$

The function given by $$R(x)=11.74 x^{0.25}$$ can be used to approximate the maximum range, \(R(x)\), in miles, of ARSR-3 surveillance radar with a peak power of \(x\) watts. a) Determine the maximum radar range when the peak power is 40,000 watts, 50,000 watts, and 60,000 watts. b) Graph the function.

An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones. The humerus is the bone from the elbow to the shoulder. Let \(x=\) the length of the humerus, in centimeters. Then the height, in centimeters, of a male with a humerus of length \(x\) is given by $$ M(x)=2.89 x+70.64 $$. The height, in centimeters, of a female with a humerus of length \(x\) is given by $$ F(x)=2.75 x+71.48$$ A \(26-\mathrm{cm}\) humerus was uncovered in some ruins. a) If we assume it was from a male, how tall was he? b) If we assume it was from a female, how tall was she?
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