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Chapter 1: Problem 12

Find \(f(3)\), if it exists, for each of the following functions: a. \(f(x)=(x-3)^{2}\) b. \(f(x)=\frac{1}{x}\) c. \(f(x)=\frac{x+1}{x-3}\) d. \(f(x)=\frac{2 x}{x-1}\) Determine the domain for each function.

Short Answer

Expert verified
a. 0, Domain: \[ (-\infty, \infty) \]; b. \( \frac{1}{3} \), Domain: \[ (-\infty, 0) \cup (0, \infty) \]; c. Undefined, Domain: \[ (-\infty, 3) \cup (3, \infty) \]; d. 3, Domain: \[ (-\infty, 1) \cup (1, \infty) \].

Step by step solution

01

- Evaluate function a at x = 3

Given the function \( f(x) = (x-3)^2 \), substitute \( x = 3 \): \[ f(3) = (3-3)^2 = 0^2 = 0 \]
02

- Determine the domain for function a

The function \( f(x) = (x-3)^2 \) is defined for all real numbers. Therefore, the domain is \[ (-\infty, \infty) \].
03

- Evaluate function b at x = 3

Given the function \( f(x) = \frac{1}{x} \), substitute \( x = 3 \): \[ f(3) = \frac{1}{3} \]
04

- Determine the domain for function b

The function \( f(x) = \frac{1}{x} \) is undefined when \( x = 0 \). Therefore, the domain is \[ (-\infty, 0) \cup (0, \infty) \].
05

- Evaluate function c at x = 3

Given the function \( f(x) = \frac{x+1}{x-3} \), substitute \( x = 3 \): \[ f(3) = \frac{3+1}{3-3} = \frac{4}{0} \] This expression is undefined.
06

- Determine the domain for function c

The function \( f(x) = \frac{x+1}{x-3} \) is undefined when \( x = 3 \). Therefore, the domain is \[ (-\infty, 3) \cup (3, \infty) \].
07

- Evaluate function d at x = 3

Given the function \( f(x) = \frac{2x}{x-1} \), substitute \( x = 3 \): \[ f(3) = \frac{2(3)}{3-1} = \frac{6}{2} = 3 \]
08

- Determine the domain for function d

The function \( f(x) = \frac{2x}{x-1} \) is undefined when \( x = 1 \). Therefore, the domain is \[ (-\infty, 1) \cup (1, \infty) \].

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Key Concepts

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

Domain Determination
When we talk about the domain of a function, we're referring to all the possible input values (usually denoted as 'x') that will give us a valid output. This is super important because not every function accepts every x value. For example, a function with a denominator that can become zero (like \( \frac{1}{x} \)) can鈥檛 take x values that make the denominator zero because division by zero is not defined.
To determine the domain:
  • Identify any values that could make the function undefined (like zeroing a denominator or taking the square root of a negative number).
  • Exclude those values from the set of all real numbers.
    So, for our functions:
    • Function a (\(f(x)=(x-3)^2\)): Defined for all real numbers.
    • Function b (\(f(x)=\frac{1}{x}\)): Undefined when x = 0.
    • Function c (\(f(x)=\frac{x+1}{x-3}\)): Undefined when x = 3.
    • Function d (\(f(x)=\frac{2x}{x-1}\)): Undefined when x = 1.
Undefined Expression
An expression becomes undefined when we try to perform an operation that doesn't make sense mathematically. This most commonly happens with:
  • Division by zero: For example, in \(f(x)=\frac{x+1}{x-3} \), substituting x = 3 leads to \( \frac{4}{0} \), an undefined operation.
  • Square roots of negative numbers: Such cases are not prevalent in this specific exercise but are critical to mention.
    To spot undefined expressions, look for cases where the denominator in a fraction equals zero or other problematic operations. Always remember where these undefined points are as they define parts of the domain.
Function Substitution
Function substitution involves replacing the variable with a specific value to find the output. Here鈥檚 how you do it:
  • Identify the function and the given value, say x = 3.
  • Replace all instances of x in the function with 3.
  • Simplify to find the result.
    Examples from the exercise:
    • For \(f(x)=(x-3)^2 \), substitute x = 3: \( f(3) = (3-3)^2 = 0 \).
    • For \( \frac{2x}{x-1} \), substitute x = 3: \( f(3) = \frac{6}{2} = 3 \).
      Make sure the substitution doesn't lead you to an undefined expression.
Real Numbers
Real numbers cover all the numbers on the number line, including both rational (like fractions and integers) and irrational numbers (like 鈭�2 or 蟺). Understanding real numbers is key to grasping domains because:
  • Most functions are defined on real numbers unless they involve actions that aren't allowed (like dividing by zero or taking the square root of negatives).
  • The domain of a function can often be thought of as 'all real numbers except certain points.'.
    For instance, the domain of \( f(x)=\frac{1}{x} \) is all real numbers unless x = 0.

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Most popular questions from this chapter

(Use of calculator or other technology recommended.) Use the following table to generate an estimate of the mean age of the U.S. population. Show your work. (Hint: Replace each age interval with an age approximately in the middle of the interval.) $$ \begin{aligned} &\text { Ages }\\\ &\text { of US. Population in } 2004\\\ &\begin{array}{lr} \hline \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Population } \\ \text { (thousands) } \end{array} \\ \hline \text { Under 10 } & 39,677 \\ 10-19 & 41,875 \\ 20-29 & 40,532 \\ 30-39 & 41,532 \\ 40-49 & 45,179 \\ 50-59 & 35,986 \\ 60-74 & 31,052 \\ 75-84 & 12,971 \\ 85 \text { and over } & 4,860 \\ \hline \text { Total } & 293,655 \\ \hline \end{array} \end{aligned} $$

Assume that for persons who earn less than \(\$ 20,000\) a year, income tax is \(16 \%\) of their income. a. Generate a formula that describes income tax in terms of income for people earning less than \(\$ 20,000\) a year. b. What are you treating as the independent variable? The dependent variable? c. Does your formula represent a function? Explain. d. If it is a function, what is the domain? The range?

(Graphing program required.) Using technology, graph each function over the intervals [-6,6] for \(x\) and [-20,20] for \(y\). $$ y_{1}=x^{2}-3 x+2 \quad y_{2}=0.5 x^{3}-2 x-1 $$ For each function, a. Estimate the maximum value of \(y\) on each interval. b. Estimate the minimum value of \(y\) on each interval.

If \(f(x)=(2 x-1)^{2},\) evaluate \(f(0), f(1),\) and \(f(-2)\)

(Graphing program required.) Use technology to graph each function. Then approximate the \(x\) intervals where the function is concave up, and then where it is concave down. a. \(h(x)=x^{4}\) b. \(k(x)=x^{4}-24 x+50\) (Hint: Use an interval of [-5,5] for \(x\) and [0,200] for \(y .)\)
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