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Chapter 2: Problem 15

In Exercises \(7-16,\) for the given functions \(f\) and \(g,\) find each composite function and identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \(\left(\frac{f}{g}\right)(x)\) $$f(x)=|x| ; g(x)=\frac{1}{2 x+5}$$

Short Answer

Expert verified
The composite functions and their domains are: \n(a) (f+g)(x) = \(|x| + \frac{1}{2x+5}\), domain: all real numbers except -5/2 \n(b) (f-g)(x) = \(|x| - \frac{1}{2x+5}\), domain: all real numbers except -5/2 \n(c) (f.g)(x) = \(|x| \times \frac{1}{2x+5}\), domain: all real numbers except -5/2 \n(d) \((\frac{f}{g})(x) = 2x|x|+5|x|\), domain: all real numbers except -5/2.

Step by step solution

01

Calculate (f+g)(x)

Find the sum of function \(f(x)\) and function \(g(x)\) denoted as \((f+g)(x)\). Simply add the two functions together to get \(f(x) + g(x) = |x| + \frac{1}{2x+5}\). The domain for this composite function is all real numbers, except -5/2 to prevent division by zero in \(g(x)\).
02

Calculate (f-g)(x)

Find the difference of function \(f(x)\) and function \(g(x)\) denoted as \((f-g)(x)\). Simply subtract \(g(x)\) from \(f(x)\) to get \(f(x) - g(x) = |x| - \frac{1}{2x+5}\). As before, the domain for this function is all real numbers except -5/2.
03

Calculate (f.g)(x)

Find the product of function \(f(x)\) and function \(g(x)\) denoted as \(fg(x)\). Simply multiply the two functions to get \(f(x)g(x) = |x| \times \frac{1}{2x+5}\). As before, the domain is all real numbers except -5/2.
04

Calculate \((\frac{f}{g})(x)\)

Find the quotient of function \(f(x)\) and function \(g(x)\) denoted as \((f/g)(x)\). Simply divide \(f(x)\) by \(g(x)\) to get \((f/g)(x) = \frac{|x|}{\frac{1}{2x+5}} = 2x|x|+5|x|\). The domain for this composite function is all real numbers, except -5/2 as well.

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Key Concepts

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

Function Composition
Understanding function composition is essential in calculus and algebra. It refers to the combining of two functions where the output of one function becomes the input of another. To visualize this, imagine functions as machines that process input values and produce outputs. When we compose functions, we're essentially feeding the output of one machine directly into another.

For instance, given two functions, such as the example we have with function f(x) being |x| and function g(x) being \( \frac{1}{2x+5} \), when we compose them (like \(f+g\), \(f-g\), \(f \cdot g\), or \(\frac{f}{g}\)), we're performing operations combining the two functions' outputs according to basic arithmetic operations.
Domain of a Function
The domain of a function represents all the possible input values (typically 'x' values) that the function can legitimately accept without causing undefined expressions or contradictions in mathematics. Calculating the domain is crucial for understanding the limits within which a function operates.

For our example functions, the domain of each individual function must be considered when they are composed. For both f(x) and g(x), as well as for their compositions, we must exclude the value that makes the denominator of g(x) equal to zero, which would be -5/2 in this case. Any other real number would be acceptable as input.
Piecewise Functions
Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the overall domain. Although our example does not feature a classic piecewise function, understanding them helps when dealing with complex functions that behave differently across their domain.

The concept also helps when considering absolute value, such as function f(x) = |x| in our example. While not traditionally presented as piecewise, |x| can be thought of as a piecewise function because it behaves as 'x' for positive numbers and '鈭抶' for negative numbers.
Rational Expressions
A rational expression is a fraction composed of polynomials in the numerator and denominator. In calculus and algebra, it is important to understand how to work with them because they occur frequently in equations and functions.

Taking our example function g(x), which is a rational expression, we can determine that its domain excludes values that make the denominator zero. Hence, operations involving rational expressions, such as forming composite functions, require careful evaluation of the domain to avoid division by zero, as seen in all parts of the given exercise.

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Most popular questions from this chapter

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$G(x)=-6 x+x^{2}+5$$

The conversion of temperature units from degrees Fahrenheit to degrees Celsius is given by the equation \(C(x)=\frac{5}{9}(x-32),\) where \(x\) is given in degrees Fahrenheit. Let \(T(x)=70+4 x\) denote the temperature, in degrees Fahrenheit, in Phoenix, Arizona, on a typical July day, where \(x\) is the number of hours after 6 A.M. Assume the temperature model holds until 4 P.M. of the same day. Find \((C \circ T)(x)\) and explain what it represents.

The perimeter of a square is \(P(s)=4 s\) where \(s\) is the length of a side in inches. The function \(C(x)=2.54 x\) takes \(x\) inches as input and outputs the equivalent result in centimeters. Find \((C \circ P)(s)\) and explain what it represents.

The height of a ball after being dropped from the roof of a 200 -foot-tall building is given by \(h(t)=-16 t^{2}+200,\) where \(t\) is the time in seconds since the ball was dropped, and \(h(t)\) is in feet. (a) When will the ball be 100 feet above the ground? (b) When will the ball reach the ground? (c) For what values of \(t\) does this problem make sense (from a physical standpoint)?

When designing buildings, engineers must pay careful attention to how different factors affect the load a structure can bear. The following table gives the load in terms of the weight of concrete that can be borne when threaded rod anchors of various diameters are used to form joints. $$\begin{array}{cc} \text { Diameter (in.) } & \text { Load (1b) } \\ \hline 0.3750 & 2105 \\ 0.5000 & 3750 \\ 0.6250 & 5875 \\ 0.7500 & 8460 \\ 0.8750 & 11,500 \end{array}$$ (a) Examine the table and explain why the relationship between the diameter and the load is not linear. (b) The function $$f(x)=14,926 x^{2}+148 x-51$$ gives the load (in pounds of concrete) that can be borne when rod anchors of diameter \(x\) (in inches) are employed. Use this function to determine the load for an anchor with a diameter of 0.8 inch. (c) since the rods are drilled into the concrete, the manufacturer's specifications sheet gives the load in terms of the diameter of the drill bit. This diameter is always 0.125 inch larger than the diameter of the anchor. Write the function in part (b) in terms of the diameter of the drill bit. The loads for the drill bits will be the same as the loads for the corresponding anchors. (Hint: Examine the table of values and see if you can present the table in terms of the diameter of the drill bit.)
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