1.6E: Combinations and Compositions of Functions (Exercises)

SECTION 1.6 EXERCISES

1. Use the table of values to evaluate each expression

The inputs and outputs of functions f and g

x	0	1	2	3	4	5
\(f(x)\)	7	2	1	0	12	4
\(g(x)\)	5	3	2	1	4	0

1. \((f + g)(1)\)
2. \((f \cdot g)(2)\)
3. \((\dfrac{f}{g})(3)\)
4. Write a table of values that defines the function \((f + g)(x)\)
5. Write a table of values that defines the function \((f \cdot g)(x)\)
6. \((f \circ g)(1)\)
7. \((f \circ g)(2)\)
8. \((f \circ g)(4)\)
9. \((g \circ f)(1)\)
10. \((g \circ f)(3)\)
11. \((g \circ f)(4)\)
12. Write a table of values that defines the function \((f \circ g)(x)\)

2. Use the table of values to evaluate each expression

The inputs and outputs of functions k and h

x	-2	-1	0	1	2	3
\(h(x)\)	3	1	0	-1	-2	1
\(k(x)\)	7	3	2	3	5	-1

1. \((h - k)(-2)\)
2. \((h \cdot k)(1)\)
3. \((\dfrac{h}{k})(3)\)
4. \((\dfrac{k}{h})(0)\)
5. Write a table of values that defines the function \((h - k)(x)\)
6. Write a table of values that defines the function\((\dfrac{k}{h})(x)\)
7. \((k \circ h)(1)\)
8. \((k \circ h)(2)\)
9. \((h \circ k)(1)\)
10. \((h \circ k)(2)\)
11. Write a table of values that defines the function \((h \circ k)(x)\)

3. Use the graphs to evaluate the expressions below.

1. \((f + g)(2)\)
2. \((f \cdot g)(1)\)
3. \((\dfrac{f}{g})(2)\)
4. \((f \circ g)(1)\)
5. \((f \circ g)(2)\)
6. \((g \circ f)(1)\)
7. \((g \circ f)(2)\)

4. Use the graphs to evaluate the expressions below.

1. \((h - k)(5)\)
2. \((h \cdot k)(1)\)
3. \((\dfrac{h}{k})(3)\)
4. \((\dfrac{k}{h})(0)\)
5. \((k \circ h)(1)\)
6. \((k \circ h)(2)\)
7. \((h \circ k)(0)\)
8. \((h \circ k)(1)\)

5. Let \(f(x) = 4x + 8\) and \(g(x) = 7 - x^{2}\)

1. Evaluate \((f + g)(1)\)
2. \((f \cdot g)(1)\)
3. \((\dfrac{f}{g})(2)\)
4. Find a formula for the sum function \(f + g\) in terms of \(x\).
5. Find a formula for the product function \(f \cdot g\) in terms of \(x\).
6. Evaluate \((f \circ g)(1)\)
7. Evaluate \((g \circ f)(1)\)
8. Evaluate \((f \circ g)(3)\)
9. Find a formula for the composition \(f \circ g\) in terms of \(x\).
10. Find a formula for the composition \(g \circ f\) in terms of \(x\).

6. Let \(f(x) = \sqrt{x + 4}\) and \( g(x) = x^{2}+5x\)

1. Evaluate \((f - g)(5)\)
2. \((f \cdot g)(1)\)
3. \((\dfrac{g}{f})(2)\)
4. Find a formula for the function \(f - g\) in terms of \(x\).
5. Find a formula for the function \(\dfrac{f}{g})\) in terms of \(x\).
6. Evaluate \((f \circ g)(1)\)
7. Evaluate \((g \circ f)(5)\)
8. Evaluate \((f \circ g)(3)\)
9. Find a formula for the composition \(f \circ g\) in terms of \(x\).
10. Find a formula for the composition \(g \circ f\) in terms of \(x\).

7. Let \(k(x) = \dfrac{1}{x + 2} \) and \( h(x) = 4x + 3\)

1. Evaluate \((k + h)(1)\)
2. \((h \cdot k)(1)\)
3. \((\dfrac{k}{h})(-2)\)
4. Find a formula for the function \(f + g\) in terms of \(x\).
5. Find a formula for the function \(f \cdot g\) in terms of \(x\).
6. Evaluate \((k \circ h)(1)\)
7. Evaluate \((h \circ k)(-1)\)
8. Evaluate \((k \circ h)(3)\)
9. Find a formula for the composition \(h \circ k\) in terms of \(x\).
10. Find a formula for the composition \(k \circ h\) in terms of \(x\).

8. Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as the composition of \(f\) and \(g\) where \(h(x)=f(g(x))\).

1. \(h(x)=(x+2)^{2}\)
2. \(h(x)=\dfrac{3}{x-5}\)
3. \(h(x)=3+\sqrt{x-2}\)

9. Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as the composition of \(f\) and \(g\) where \(h(x)=f(g(x))\).

1. \(h(x)=(x-5)^{3}\)
2. \(h(x)=\dfrac{4}{(x+2)^{2}}\)
3. \(h(x)=4+\sqrt[{3}]{x+1}\)

10. The function \(x=D(p)=-2p+300\) gives the number of items that will be demanded when the price is \(p\). The production cost, \(C=f(x) = 5x+200 \) is the cost of producing \(x\) items.

1. Find a formula for a function \(C = k(p)\) that gives the production cost \(C\) when a price \(p\) is charged.
2. What is the production cost when a price of $24 is charged.

11. The US Office of Emergency Management calculates risk by multiplying the probability of hazard by the degree of vulnerability. If a natural disaster, such as a hurricane or a wild fire, was expected to impact a certain region, scientists would perform a disaster risk assessment using probability equations to evaluate the potential liability and loss to property and even human life. Suppose the probability of hazard in \(t\) years in a particular region is given by \(H = f(t) = 0.0012t\). Suppose the degree of vulnerability of the same region is given by \(V = g(t) = 0.24(1.013)^t\).

1. Find a formula \(R=k(t)\) for the risk \(R\) to the region in \(t\) years.
2. Use your formula to calculate the risk to the region in 10 years.

12. The radius \(r\), in inches, of a spherical balloon is related to the volume, \(V\), by the function \(r=f(V)=\sqrt[{3}]{\dfrac{3V}{4\pi } }\). Air is pumped into the balloon, so the volume after \(t\) seconds is given by \(V=g(t) = 10 + 20t\).

1. Find a formula for a function \(r=k(t)\) that gives the radius \(r\) after \(t\) seconds.
2. Use your formula to find the radius after 20 after 20 seconds.

13. The number of bacteria in a refrigerated food product is given by \(N(T) = 23T^{2} - 56T + 1\), \(3 < T < 33\), where \(T\) is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by \(T(t) = 5t + 1.5\), where t is the time in hours.

1. Find a formula \(N = f(t) \)for the number of bacteria in the refrigerated food product in \(t\) hours.
2. Find the bacteria count after 4 hours

Answer

1. a) 5 b) 2 c) 0 d) & e) are shown in the tables below

x	0	1	2	3	4	5
\(y = (f + g)(x) \)	12	5	3	1	16	4

x	0	1	2	3	4	5
\(y = (f \cdot g)(x) \)	35	6	2	0	48	0

f) 0 g) 1 h) 12 i) 2 j) 5 k) undefined l) shown below in the table

x	0	1	2	3	4	5
\(y = f(g(x)) \)	4	0	1	2	12	7

3. a) 5 b) 9 c) undefined d) 2 e) 4 f) 4 g) 1

5. a) 18 b) 48 c) \( \dfrac{16}{3}\) d) \((f+g)(x) = -x^2+4x+15 \) e) \((f \cdot g)(x) = 4x^3 - 8x^2+28x+56 \) f) 32

g) -137 h) 0 i) \((f(g(x)) = -4x^2+36 \) j) \((g(f(x)) = -16x^-64x-57 \)

9. a) \(f(x) = x^3\) and \(g(x)=x-5\) b) \(f(x) = \dfrac{4}{x^2}\) and \(g(x)=x+2\) c) \(f(x) = 3 + \sqrt[3]{x}\) and \(g(x)=x+1\)

11. \(R=k(t) = (0.24(1.013)^t)( 0.0012t) = 0.000288(1.013)^t\)