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Chapter 4: Problem 13

In \(11-18 :\) a. Find \(h(x)\) when \(h(x)=g(f(x)) .\) b. What is the domain of \(h(x) ?\) c. What is the range of \(\mathrm{h}(x) ?\) d. Graph \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=x^{2}, \mathrm{g}(x)=4+x $$

Short Answer

Expert verified
a. \( h(x) = 4 + x^2 \); b. Domain: \( (-\infty, \infty) \); c. Range: \( [4, \infty) \); d. Graph: upward parabola.

Step by step solution

01

Find Expression for h(x)

Given \( h(x) = g(f(x)) \), we substitute the expressions for \( f(x) \) and \( g(x) \). First, calculate \( f(x) = x^2 \). Then, substitute this into \( g(x) \), which gives us \( g(f(x)) = 4 + f(x) \). This results in \( h(x) = 4 + x^2 \).
02

Determine Domain of h(x)

The function \( f(x) = x^2 \) is defined for all real numbers as squaring any real number is valid. Additionally, \( g(x) = 4 + x \) is also defined for all real numbers. Hence, the domain of \( h(x) \), which is \( 4 + x^2 \), includes all real numbers. Thus, the domain of \( h(x) \) is \( (-\infty, \infty) \).
03

Determine Range of h(x)

The range of \( f(x) = x^2 \) is \( [0, \infty) \) as squaring any real number gives a non-negative result. For \( g(x) = 4 + x \) where \( x = f(x) \), it adds 4 to this output, transforming the range to \( [4, \infty) \). Thus, the range of \( h(x) = 4 + x^2 \) is \( [4, \infty) \).
04

Graph h(x)

To graph \( h(x) = 4 + x^2 \), start by noting it is a parabola opening upwards, shifted 4 units up from the origin due to the constant term +4. The vertex is at \((0, 4)\), and the graph is symmetric about the y-axis. The shape is identical to \( x^2 \) but translated upwards by 4 units.

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Key Concepts

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

Domain
When we talk about the domain of a function, we are looking at the complete set of input values that the function can take. For our function, \( h(x) = 4 + x^2 \), we need to consider the domains of both \( f(x) = x^2 \) and \( g(x) = 4 + x \).
  • The function \( f(x) = x^2 \) accepts any real number as input because any number squared is a valid mathematical operation. Hence, its domain is \( (-\infty, \infty) \).
  • Likewise, \( g(x) = 4 + x \) also accepts all real numbers since you can add 4 to any real number. Thus, its domain is \( (-\infty, \infty) \).
Combining these, the domain of the composed function \( h(x) = 4 + x^2 \) is all real numbers, or \( (-\infty, \infty) \), as it inherits the domain from \( f(x) \) and \( g(x) \).
This means you can plug any real number into \( h(x) \), and it will produce a valid output.
Range
The range of a function refers to all possible output values. For \( h(x) = 4 + x^2 \), we determine this based on the behavior of \( f(x) = x^2 \) and how it affects \( g(x) = 4 + x \).
  • Starting with \( f(x) = x^2 \), for any real input \( x \), the output is always a non-negative number, resulting in a range of \( [0, \infty) \). This is because squared numbers are always positive or zero.
  • The function \( g(x) = 4 + x \) translates this output upwards by 4. So, the smallest value that \( g(x) \) can produce is 4 when \( x = 0 \), and it extends to infinity.
Therefore, the range of \( h(x) = 4 + x^2 \) starts from 4 and goes to infinity, expressed as \( [4, \infty) \). This shows that no matter the input, you can expect outputs of \( h(x) \) to always be 4 or greater.
Graphing Functions
Graphing \( h(x) = 4 + x^2 \) involves understanding how the function behaves visually on a coordinate plane.
  • It is a transformed version of the basic parabola \( y = x^2 \). The term \(+4\) indicates a vertical shift upwards.
  • The vertex of the graph is at the point \((0, 4)\) because \( h(x) \) is simply \( x^2 \) moved 4 units upwards.
  • Because the graph is a parabola, it is symmetric about the y-axis and opens upwards.
Considering these points, drawing the graph includes plotting the vertex and understanding that for every unit you move horizontally away from \( x = 0 \), the value of \( h(x) \) grows quadratically. Two main points to remember for sketching: When \( x = 1 \) or \( x = -1 \), \( h(x) = 5 \), confirming the bell shape of the parabola shifted upwards.
This visualization helps to see why the range is \([4, \infty)\), as the graph starts at y = 4 and extends infinitely up.

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Most popular questions from this chapter

In \(23-28,\) write an equation of the direct variation described. At \(9 : 00\) A.M., Christina began to add water to a swimming pool at the rate of 25 gallons per minute. When she began, the pool contained 80 gallons of water. Christina stopped adding water to the pool at \(4 : 00\) P.M. Let \(g\) be the number of gallons of water in the pool and \(t\) be the number of minutes that have past since 9\(\cdot 00\) A.M. a. Write an equation for \(g\) as a function of \(t .\) b. What is the domain of the function? c. What is the range of the function? d. Is the function one-to-one? e. Is the function an example of direct variation? Explain why or why not.

In \(7-10,\) the domain of each function is the set of real numbers. a. Sketch the graph of each function. b. What is the range of each function? $$ \mathrm{f}(x)=|4 x|+1 $$

Explain the difference between fg \((x)\) and \(f(g(x))\)

Let \(c(x)\) represent the cost of an item, \(x,\) plus sales tax and \(d(x)\) represent the cost of an item less a discount of \(\$ 10 .\) a. Write \(c(x)\) using an 8\(\%\) sales tax. b. Write \(\mathrm{d}(x)\) c. Find \(c \circ \mathrm{d}(x)\) and \(\mathrm{d} \circ \mathrm{c}(x) .\) Does \(\mathrm{c} \circ \mathrm{d}(x)=\mathrm{d} \circ \mathrm{c}(x) ?\) If not, explain the difference between the two functions. When does it makes sense to use each function? d. Which function can be used to find the amount that must be paid for an item with a \(\$ 10\) discount and 8\(\%\) tax?

In \(23-28,\) write an equation of the direct variation described. The distance in miles, \(d\) , that Mr. Spencer travels is directly proportional to the length of time in hours, \(t,\) that he travels at 35 miles per hour.
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