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TEKS
AR.3D Represent a re-
sulting function tabularly,
graphically, and symbol-
ically when functions are
combined or separated us-
ing arithmetic operations,
such as combining a 20%
discount and a 6% sales tax
on a sale to determine h ( x ),
the total sale, f ( x ) = 0.8 x ,
g ( x ) = 0.06(0.8 x ), and
h ( x ) = f ( x ) + g ( x ).
MATHEMATICAL PROCESS SPOTLIGHT
AR.1A Apply mathe-
matics to problems arising
in everyday life, society,
and the workplace.
ELPS
3F Use visual and
contextual support and
support from peers and
teachers to read grade-ap-
propriate content area
text, enhance and confirm
understanding, and devel-
op vocabulary, grasp of
language structures, and
background knowledge
needed to comprehend
increasingly challenging
language.
VOCABULARY
function, input, output,
domain, range
MATERIALS
- graph paper
- graphing calculator
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FOCUSING QUESTION How can you generate a new function by combining
the graphs of two functions?
LEARNING OUTCOMES
■ ■ I can combine or separate two functions using graphs. ■ ■ I can represent the addition or subtraction of two functions using a graph or using symbols. ■ ■ I can apply mathematics to solve problems arising in society.
Adding and Subtracting
Functions Using Graphs
ENGAGE
A pumpkin patch uses the function f ( x ) = x (20 – x ) to estimate the cost of planting x acres of pumpkins. The function g ( x ) = 2 x (3 + 1.50 x ) represents income generat- ed by selling x acres of pumpkins. What function would represent the profit from selling x acres of pumpkins? h(x) = 4x^2 − 14x
EXPLORE
Cool Cars, Inc., is a car rental company. For a compact car, they charge $25 per day for the rental. Local taxes and facility fees are an additional $8.50 per day and $7. for the rental period.
1. Write a function, f ( x ), which describes the amount of the rental fee for a rental period of x days. f(x) = 25x 2. Write a function, g ( x ), which describes the amount of local taxes and facility fees for a rental period of x days. g(x) = 8.50x + 7 3. Graph both f ( x ) and g ( x ) on the same grid. See margin.
386 C H A P T E R 4 : F U N C T I O N O P E R AT I O N S
4. Answers may vary.
Possible answer:
The graph of h(x) will be
a combination of f(x) and
g(x) together. The graph
should be a line that is
steeper than f(x) or g(x)
because the two costs
have been combined into a
larger cost.
6. Answers may vary.
Possible answer: My
prediction was close to the
line created by the points
for h(x), but the actual
graph of h(x) is steeper
than what I predicted.
386 C H A P T E R 4 : F U N C T I O N O P E R AT I O N S
4. The function, h ( x ), represents the total cost of the rental for x days. Let h ( x ) = f ( x ) + g ( x ). Sketch a prediction of the graph of h ( x ) on the graph as f ( x ) and g ( x ). Explain your prediction. See margin. 5. Create a table of values for x , f ( x ), g ( x ), and h ( x ).
x
RENTAL FEE (DOLLARS) f ( x )
TAXES AND FEES (DOLLARS) g ( x )
TOTAL COST (DOLLARS) h ( x ) = f ( x ) + g ( x )
0 0 7.00 7. 1 25.00 15.50 40. 2 50.00 24.00 74. 3 75.00 32.50 107. 4 100.00 41.00 141. 5 125.00 49.50 174. 6 150.00 58.00 208. 7 175.00 66.50 241.
6. Plot the points for ( x , h ( x )) onto the graph and connect them with a line or curve. How does the line or curve compare to your prediction? See margin. 7. Plot the points for (1, f (1)) and (1, g (1)). How do the y -coordinates of these points relate to the y -coordinate of (1, h (1))? f(1) + g(1) = 25 + 15.50 = 40.50 = h(1) 8. Plot the remaining points for ( x , f ( x )) and ( x , g ( x )). How do these points relate to ( x , h ( x )) when the x -coordinates are the same? See margin. 9. Use the graph to write an equation for h ( x ). h(x) = f(x) + g(x) = 25x + (8.50x + 7) = 33.50x + 7
REFLECT
■ ■ How is a third graph,h(x), generated by adding two graphs,f(x) and g(x), together? For each x-value, add the function values for f(x) and g(x) together. The sum is the function value for h(x).
■ ■ How could you create a fourth function,p(x), by subtractingf(x) fromg(x)? For each x-value, subtract the function value for f(x) from the function value for g(x). The difference is the function value for p(x).
388 C H A P T E R 4 : F U N C T I O N O P E R AT I O N S
388 C H A P T E R 4 : F U N C T I O N O P E R AT I O N S
The difference between the number of hours Ginny works in 4 weeks and the number of hours Franklin works in 4 weeks, h (4), is the difference between these two numbers, or 160 − 100 = 60 hours. This difference is represented by the point (4, 60) on the graph of h ( x ).
If you repeat this process for additional x-values, then you will generate the graph of h ( x ).
Subtracting functions using graphs means to subtract the function values for two existing functions in order to generate function values for a new third function.
h ( x ) = g ( x ) − f ( x )
ADDING AND SUBTRACTING FUNCTIONS USING GRAPHS
Graphs can be used to combine (add) or separate (subtract) functions. ■ ■ To add functions, identify the function values ( y-values) for the same input value (x-value) and add them to- gether. Plot the sum (x,f(x) +g(x)). Repeat this pro- cess for additionalx-values. ■ ■ To subtract functions, identify the function values (y-values) for the same input value (x-value) and subtract. Plot the difference (x,g(x) − f(x)). Repeat this process for additionalx-values.
EXAMPLE 1
Use the graph of the functions f ( x ) = – ( x – 3) 2 + 1 and g ( x ) = x – 4 to graph and write the equation of the combined function h ( x ) = f ( x ) + g ( x ).
ADDITIONAL EXAMPLES
Use the graph of the
function f ( x ) and g ( x ) below
to graph and write the
equation of the combined
function h ( x ) = f ( x ) + g ( x ).
1. f ( x ) = x + 2
g ( x ) = ( x + 1)^2 – 3
h(x) = x^2 + 3x
2. f ( x ) = ( x – 3)^2 + 1
g ( x ) = ( x – 1)^3 – 4
h(x) = x^3 – 2x^2 – 3x + 5
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STEP 1 Determine points with the same x -values on both graphs.
x f ( x ) g ( x ) 0 – 8 – 4 1 – 3 – 3 2 0 – 2 3 1 – 1 4 0 0 5 – 3 1 6 – 8 2
STEP 2 Graph the points ( x , h ( x )) and draw the curve through the points.
x (^) f ( x ) g ( x ) h ( x ) 0 – 8 – 4 – 12 1 – 3 – 3 – 6 2 0 – 2 – 2 3 1 – 1 0 4 0 0 0 5 – 3 1 – 2 6 – 8 2 – 6
STEP 3 Use either the patterns in the table of points for h ( x ) or the
equations of the functions f ( x ) and g ( x ) to write a function for h ( x ).
h ( x ) = f ( x ) + g ( x ) h ( x ) = [– ( x – 3)^2 + 1] + [ x – 4] h ( x ) = [– ( x^2 – 6 x + 9) + 1] + x – 4 h ( x ) = [– x^2 + 6 x – 9 + 1] + x – 4 h ( x ) = – x^2 + 6 x – 8 + x – 4 h ( x ) = – x^2 + 6 x + x – 8 – 4 h ( x ) = – x^2 + 7 x – 12
YOU TRY IT! #2 ANSWER:
w(x) = 9x^2 – 6
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STEP 2 Graph the points ( x , h ( x )) and draw the curve through the points.
x f ( x ) g ( x ) h ( x )
- 3 – 8 – 2 – 6
- 2 – 1 – 2 1
- 1 0 – 2 2 0 1 – 2 3 1 8 – 2 10
STEP 3 Use the graph and your knowledge of function transformations
to write a function for h ( x ).
The shape of h(x) is the same as the cubic parent function, y = x^3 , if it were translated 1 unit to the left and 2 units up. Therefore, h(x) = (x + 1)^3 + 2 = x^3 + 3x^2 + 3x + 3.
YOU TRY IT! #
Use the graph of the functions u ( x ) = 6 x + 3 and v ( x ) = (3 x + 1) 2 – 4 to graph and write the equation of the combined function w ( x ) = v ( x ) – u ( x ). See margin.
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QUESTIONING
STRATEGY
For Example 3 and YOU
TRY IT! #3 , ask students
to first assess whether the
functions should be added
together or separated with
subtraction. Ask students
to explain why they made
the choice they did.
ADDITIONAL EXAMPLE
Joel regularly purchases
games, music, and videos
on his smart phone. He
spends about $12 each
month. His parents gave
him $50 for Christmas, and
they give him a $15 per
month allowance to use as
he pleases. The function
p ( x ) = 12 x can be used to
represent the purchases the
Joel makes over x months.
The function m ( x ) = 15 x + 50
can be used to represent
the discretionary funds that
Joel has from his parents
over x months. The graph
represents this situation.
Use the graph to graph
and write a function
j ( x ) to represent Joel’s
monthly available funds
after x months.
j(x) = 3x + 50
392 C H A P T E R 4 : F U N C T I O N O P E R AT I O N S
EXAMPLE 3
The graph shows projections of population, in thou- sands of people, for two neighboring communities that are growing together over the next x years and plan to incorporate as a city. The population of one community is projected to grow according to the function f ( x ) = 1 – 3 x + 4. The population of the other com- munity is projected to grow according to the function g(x) = — 121 x^2 + 3.
Graph and write a function h ( x ) to represent the total projected population growth, in thousands of people, over the next x years of the two communities that plan to incorporate as a city.
STEP 1 Determine points with the same x -values on both graphs.
x f ( x ) g ( x ) 0 4 3 6 6 6 12 8 15
STEP 2 Since h ( x ) will represent the total projected population,
h ( x ) = f ( x ) + g ( x ). Graph the points ( x , h ( x )) and draw the curve
through the points.
x f ( x ) g ( x ) h ( x ) = f ( x ) + g ( x ) 0 4 3 7 6 6 6 12 12 8 15 23
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PRACTICE/HOMEWORK
Use the functions shown to answer questions 1 – 5.
p ( x ) = 3( x + 1) – 6 q ( x ) = -2( x – 3) + 4 r ( x ) = p ( x ) + q ( x ) s ( x ) = p ( x ) – q ( x )
1. Complete the table shown for specific x values for p ( x ), q ( x ), r ( x ), and s ( x ).
x p ( x ) q ( x ) r ( x ) s ( x ) -2 -9 14 5 - -1 -6 12 6 - 0 -3 10 7 - 1 0 8 8 - 2 3 6 9 - 3 6 4 10 2 4 9 2 11 7 5 12 0 12 12
2. Sketch a graph of the functions p ( x ), q ( x ), and r ( x ). See margin. 3. Write the equation of the combined function r ( x ) = p ( x ) + q ( x ). r(x) = x + 7 4. Sketch a graph of the functions p ( x ), q ( x ), and s ( x ). See margin. 5. Write the equation of the separated function s ( x ) = p ( x ) – q ( x ). s(x) = 5x − 13
Use the functions shown to answer questions 6 – 10. f ( x ) = ( x + 2) 2 – 4 g ( x ) = ( x – 2) + 6 h ( x ) = f ( x ) + g ( x ) k ( x ) = f ( x ) – g ( x )
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6. Complete the table shown for specific x values for f ( x ), g ( x ), h ( x ), and k ( x ).
x f ( x ) g ( x ) h ( x ) k ( x ) -4 0 0 0 0 -3 -3 1 -2 - -2 -4 2 -2 - -1 -3 3 0 - 0 0 4 4 - 1 5 5 10 0 2 12 6 18 6 3 21 7 28 14
7. Sketch a graph of the functions f ( x ), g ( x ), and h ( x ). See margin. 8. Write the equation of the combined function h ( x ) = f ( x ) + g ( x ). h(x) = x^2 + 5x + 4 9. Sketch a graph of the functions f ( x ), g ( x ), and k ( x ). See margin. 10. Write the equation of the separated function k ( x ) = f ( x ) − g ( x ). k(x) = x^2 + 3x – 4
Use the functions shown to answer questions 11 – 15. p ( x ) = 8(0.5x + 1) 3 + 1 q ( x ) = -7( x – 3) r ( x ) = p ( x ) + q ( x ) s ( x ) = p ( x ) − q ( x )
11. Complete the table shown for specific x values for p ( x ), q ( x ), r ( x ), and s ( x ).
x p ( x ) q ( x ) r ( x ) s ( x ) -1 2 28 30 - 0 9 21 30 - 1 28 14 42 14 2 65 7 72 58 3 126 0 126 126 4 217 -7 210 224 5 344 -14 330 358 6 513 -21 492 534
