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Chapter 3: Problem 11

Find the domain and range of the given functions. explain your answers. $$H(h)=2 h+\sqrt{h}+1$$

Short Answer

Expert verified
Domain: \( h \geq 0 \); Range: \([1, \infty)\).

Step by step solution

01

Understand the Function

The function is given by \( H(h) = 2h + \sqrt{h} + 1 \). It consists of a linear component \( 2h + 1 \) and a square root component \( \sqrt{h} \).
02

Determine the Domain

To find the domain, identify the values of \( h \) for which the square root is defined. The expression \( \sqrt{h} \) is only defined for \( h \geq 0 \). Therefore, the domain of the function \( H(h) \) is all real numbers \( h \) such that \( h \geq 0 \).
03

Determine the Range

To find the range, analyze the potential values that \( H(h) \) can take as \( h \) varies over its domain. As \( h \) increases, both \( 2h \) and \( \sqrt{h} \) will increase. The smallest value of the function occurs at \( h = 0 \), resulting in \( H(0) = 2 \times 0 + \sqrt{0} + 1 = 1 \). The function will continue to increase without bound as \( h \) increases, suggesting the range is \([1, \infty)\).

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Key Concepts

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

Linear Function
A linear function represents a straight line when graphed and is expressed in the form of \( f(x) = ax + b \), where \( a \) and \( b \) are constants.
  • The slope of the line is represented by \( a \).
  • The y-intercept, which is where the line crosses the y-axis, is \( b \).
Linear functions have specific characteristics:
  • They grow or decrease steadily based on their slope.
  • Their domains and ranges generally include all real numbers unless specified otherwise鈥攊t means \( x \) and \( f(x) \) can take any real number.
  • They are simple and predictable.
In the expression \( H(h) = 2h + \sqrt{h} + 1 \), the \( 2h + 1 \) part represents the linear component. It suggests that as \( h \) increases, this segment's value changes directly in proportion to \( h \). This is typical of linear functions.
Square Root Function
A square root function is characterized by its defining feature: the square root symbol (\( \sqrt{} \)).
It appears as \( f(x) = \sqrt{x} \), introducing some unique properties:
  • The square root of a number is only defined for non-negative values of \( x \) due to the restriction of not having real square roots for negative numbers.
  • Its graph forms a curve that starts at the origin (0,0) and increases to the right in a non-linear fashion.
In the function \( H(h) = 2h + \sqrt{h} + 1 \), \( \sqrt{h} \) represents the square root portion, and thus affects the domain directly by requiring \( h \geq 0 \). This ensures that the function is well-defined, keeping outputs real.
Function Properties
Understanding the properties of functions like domain and range helps to analyze how functions behave.
  • **Domain**: Refers to all the possible input values (x-values) that a function can accept. For \( H(h) \), the domain is constrained by the square root to \( h \geq 0 \).
  • **Range**: Consists of all possible output values (y-values) a function can produce. Analyzing \( H(h) \), as \( h \) begins at 0 and grows, the smallest output is \( H(0) = 1 \), with the range being \([1, \infty)\).
By understanding these properties, we can clearly determine how the function behaves and predict its behavior over intervals, providing deeper insight into its overall characteristics.
Real Numbers
Real numbers are the set of numbers that include both rational and irrational numbers, effectively covering any number on the number line.
  • They include integers, fractions, and non-repeating decimal numbers.
  • Rational numbers are those that can be expressed as a fraction of two integers.
  • Irrational numbers, like \( \pi \) or \( \sqrt{2} \), cannot be expressed as simple fractions.
In the context of the function \( H(h) = 2h + \sqrt{h} + 1 \), both the domain and range mostly involve real numbers. The function's restrictions mean it's dealing specifically with non-negative real numbers due to the presence of the square root, emphasizing its crucial nature in defining allowable values for \( h \).

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Most popular questions from this chapter

Graph the indicated functions. The voltage \(V\) across a capacitor in a certain electric circuit for a \(2-s\) interval is \(V=2 t\) during the first second and \(V=4-2 t\) during the second second. Here, \(t\) is the time (in s). Plot \(V\) as a function of \(t\).

Use the following table, which gives the valve lift \(L\) (in mm) of a certain cam as a function of the angle \(\theta\) (in degrees ) through which the cam is turned. Plot the values. Find the indicated values by reading the graph. $$\begin{array}{l|l|l|l|l|l|l|l|l} \theta\left(^{\circ}\right) & 0 & 20 & 40 & 60 & 80 & 100 & 120 & 140 \\ \hline L(\mathrm{mm}) & 0 & 1.2 & 2.3 & 3.3 & 3.8 & 3.0 & 1.6 & 0 \end{array}$$ For \(L=2.0 \mathrm{mm},\) find \(\theta.\)

Graph the indicated functions. On a taxable income of \(x\) dollars, a certain city's income \(\operatorname{tax} T\) is defined as \(T=0.02 x\) if \(020,000 .\) Graph \(T=f(x)\) for \(0 \leq x<100,000\).

Graph the indicated functions. A land developer is considering several options of dividing a large tract into rectangular building lots, many of which would have perimeters of \(200 \mathrm{m}\). For these, the minimum width would be \(30 \mathrm{m}\) and the maximum width would be 70 m. Express the areas \(A\) of these lots as a function of their widths \(w\) and plot the graph.

Graph the given functions. $$y=6-2 x^{2}$$
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