Wize High School Algebra II Textbook (Common Core) > Combination of Functions

Sum & Difference of Functions

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Sum & Difference
Functions can be combined through addition and subtraction. 


Let f(x)f(x)f(x) and g(x)g(x)g(x) be two continuous functions defined on the interval (−∞,∞)(-\infin,\infin)(−∞,∞). 
The sum of 2 functions, f(x)f(x)f(x) and g(x)g(x)g(x), can be expressed as f(x)+g(x)\boxed{f(x)+g(x)}f(x)+g(x)​. 
The difference of 2 functions, f(x)f(x)f(x) and g(x)g(x)g(x), can be expressed as f(x)−g(x)\boxed{f(x)-g(x)}f(x)−g(x)​. 
Wize Tip
Both functions must be defined at a point for the combination to be defined.

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Example

Let f(x)=x−2f(x)=x-2f(x)=x−2 and g(x)=x+1g(x)=x+1g(x)=x+1. Find:
f(x)+g(x)f(x)+g(x)f(x)+g(x)
f(x)−g(x)f(x)-g(x)f(x)−g(x)

1.f(x)+g(x)=(x−2)+(x+1)=2x−12.f(x)−g(x)=(x−2)−(x+1)=−3\begin{array}{rrcl}
1.&f(x)+g(x)&=&(x-2)+(x+1)\\\\
&&=&2x-1\\\\\\
2.&f(x)-g(x)&=&(x-2)-(x+1)\\\\
&&=&-3
\end{array}1.2.​f(x)+g(x)f(x)−g(x)​====​(x−2)+(x+1)2x−1(x−2)−(x+1)−3​

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Example: Sum & Difference of Functions

Given f(x)=2x+3f(x)=2x+3f(x)=2x+3 and g(x)=x2−1g(x)=x^2-1g(x)=x2−1, determine:
f(x)+g(x)f(x)+g(x)f(x)+g(x)
Algebraically
Graphically

1. f(x)+g(x):f(x)+g(x):f(x)+g(x):                                                                        
     a. f(x)+g(x)=(2x+3)+(x2−1)=x2+2x+2=(x2+2x+1)+2−1=(x+1)2+1\begin{array}{rcl}
f(x)+g(x)&=&(2x+3)+(x^2-1)\\\\
&=&x^2+2x+2\\\\
&=&(x^2+2x+1)+2-1\\\\
&=&(x+1)^2+1
\end{array}f(x)+g(x)​====​(2x+3)+(x2−1)x2+2x+2(x2+2x+1)+2−1(x+1)2+1​  

      b. 
  

xf(x)g(x)f(x)+g(x)−2−132−110103−12150527310\begin{array}{c|c|c|c}
x&f(x)&g(x)&f(x)+g(x)\\\hline
-2&-1&3&2\\\hline
-1&1&0&1\\\hline
0&3&-1&2\\\hline
1&5&0&5\\\hline
2&7&3&10
\end{array}x−2−1012​f(x)−11357​g(x)30−103​f(x)+g(x)212510​​

  
The solution is f(x)+g(x)=(x+1)2+1f(x)+g(x)=(x+1)^2+1f(x)+g(x)=(x+1)2+1

Practice: Sum & Difference of Functions
Let f(x)=x+8f(x)=x+8f(x)=x+8 and g(x)=9x2−1g(x)=9x^2-1g(x)=9x2−1. Determine the following:

f(x) + g(x)

f(x) - g(x)

g(x) - f(x)

f(1)-g(2)

g(-1)+f(0)^2

I don't know

Practice: Sum & Difference of Functions
If m(x)=13x−4m(x)=\dfrac{1}{3x-4}m(x)=3x−41​ and n(x)=1x+1n(x)=\dfrac{1}{x+1}n(x)=x+11​, determine:

m(x)+n(x)

B. The domain of

m(x)+n(x): x\neq{}

(smaller answer)

C. The domain of

m(x)+n(x): x\neq{}

(larger answer)

m(0)+n(-2)

m(1)+n(1)

I don't know

Practice: Sum & Difference of Functions
Maisy went grocery shopping at Fresh-co and used her $5\$5$5 off coupon. She then went to London Drugs and used her 25% coupon.

A. What is the function that represents the total amount Maisy spent?

B. What is the total amount Maisy spent, given that Maisy bought items that regularly cost $80 at Fresh-co and at London Drugs. Ignore units.

I don't know

Extra Practice

Sum & Difference of Functions

Practice: Sum & Difference of Functions
Andrew is purchasing a plane ticket to Italy and he used the points that allow him $350\$350$350 off the total of the ticket. He also gets 12%12\%12% off the total of his baggage. 

Sum & Difference of Functions

Practice: Sum & Difference of Functions
Lilianna went shoe shopping at Aldo and used her $20\$20$20 off coupon. She then went to Urban Rack and used her 15%15\%15% coupon. 

Sum & Difference of Functions

Practice: Sum & Difference of Functions
If m(x)=116−x2m(x)=\dfrac{1}{16-x^2}m(x)=16−x21​ and n(x)=1x+4n(x)=\dfrac{1}{x+4}n(x)=x+41​, determine:

Sum & Difference of Functions

Practice: Sum & Difference of Functions
If a(x)=2xx2−9a(x)=\dfrac{2x}{x^2-9}a(x)=x2−92x​ and b(x)=34x−5b(x)=\dfrac{3}{4x-5}b(x)=4x−53​, determine:

Sum & Difference of Functions

Practice: Sum & Difference of Functions
Let f(x)=7x+3f(x)=7\sqrt{x}+3f(x)=7x​+3 and g(x)=2x2+5g(x)=2x^2+5g(x)=2x2+5. Determine the following:

Sum & Difference of Functions

Practice: Sum & Difference of Functions
Let f(x)=4−x2f(x)=4-x^2f(x)=4−x2 and g(x)=6x2+x−3g(x)=6x^2+x-3g(x)=6x2+x−3. Determine the following: