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

Composition of Functions

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Composition of Functions
Functions can be combined through composition. 


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 composition of 2 functions, f(x)f(x)f(x) and g(x)g(x)g(x), can be expressed as f(x)∘g(x)=f(g(x))\boxed{f(x)\circ{}g(x)=f\big(g(x)\big)}f(x)∘g(x)=f(g(x))​. 
It is read as "fff of ggg of xxx" or "fff composed of g(x)g(x)g(x)".
Substitute all values of xxx in the function f(x)f(x)f(x) with the function g(x)g(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)=x2−2x+1f(x)=x^2-2x+1f(x)=x2−2x+1 and g(x)=x+1g(x)=x+1g(x)=x+1. Find:
f(x)∘g(x)f(x)\circ{}g(x)f(x)∘g(x)
g(x)∘f(x)g(x)\circ{}f(x)g(x)∘f(x)

1.f(x)∘g(x)=f(g(x))=f(x+1)=(x+1)2−2(x+1)+1=(x2+x+1)−2x−2+1=x2−x2.g(x)∘f(x)=g(f(x))=g(x2−2x+1)=(x2−2x+1)+1=x2−2x+2\begin{array}{rrcl}
1.&f(x)\circ{}g(x)&=&f\big(g(x)\big)\\\\
&&=&f\big(x+1\big)\\\\
&&=&(x+1)^2-2(x+1)+1\\\\
&&=&(x^2+x+1)-2x-2+1\\\\
&&=&x^2-x\\\\\\
2.&g(x)\circ{}f(x)&=&g\big(f(x)\big)\\\\
&&=&g\big(x^2-2x+1\big)\\\\
&&=&(x^2-2x+1)+1\\\\
&&=&x^2-2x+2
\end{array}1.2.​f(x)∘g(x)g(x)∘f(x)​=========​f(g(x))f(x+1)(x+1)2−2(x+1)+1(x2+x+1)−2x−2+1x2−xg(f(x))g(x2−2x+1)(x2−2x+1)+1x2−2x+2​

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Example: Composition of Functions
Let f(x)=x−6f(x)=x-6f(x)=x−6 and g(x)=x2+x+4g(x)=x^2+x+4g(x)=x2+x+4. Determine:
f(g(x))f(g(x))f(g(x))
g(f(x))g(f(x))g(f(x))
f(g(0))f(g(0))f(g(0))

Part a. 
f(g(x))=(x2+x+4)−6=x2+x−2=(x+2)(x−1)\begin{array}{rcl}
f(g(x))&=&(x^2+x+4)-6\\\\
&=&x^2+x-2\\\\
&=&(x+2)(x-1)
\end{array}f(g(x))​===​(x2+x+4)−6x2+x−2(x+2)(x−1)​

Part b. 
g(f(x))=(x−6)2+(x−6)+4=(x2−12x+36)+x−2=x2−11x+34\begin{array}{rcl}
g(f(x))&=&(x-6)^2+(x-6)+4\\\\
&=&(x^2-12x+36)+x-2\\\\
&=&x^2-11x+34
\end{array}g(f(x))​===​(x−6)2+(x−6)+4(x2−12x+36)+x−2x2−11x+34​

Part c.
f(g(0))=f(02+0+4)=f(4)=4−6=−2\begin{array}{rcl}
f(g(0))&=&f(0^2+0+4)\\\\
&=&f(4)\\\\
&=&4-6\\\\
&=&-2
\end{array}f(g(0))​====​f(02+0+4)f(4)4−6−2​

Practice: Composition of Functions
Let f(x)=1x−2f(x)=\dfrac{1}{x-2}f(x)=x−21​ and g(x)=1xg(x)=\dfrac{1}{x}g(x)=x1​. Determine the following:

f(g(x))

g(f(x))

f(g(-2))

g(f(1))

I don't know

Practice: Composition of Functions
Franziska is shopping with her girlfriends at Eaton Center. She is in a store that is offering a 20% discount storewide. She also has a coupon for $10 off her purchase. The store allows you to use them in any order. 

Which is the better deal:
Using the 20%20\%20% discount first then the $10\$10$10 coupon?
Using the $10\$10$10 coupon first then the 20%20\%20% discount?

Answer

I don't know

Practice: Composition of Functions
Let f=3g−1f=3g-1f=3g−1, g=4h+6g=4h+6g=4h+6, and h=9x+10h=9x+10h=9x+10. What is f(x)?f(x)?f(x)?

f(x)=

I don't know

Extra Practice

Composition of Functions

Practice: Composition of Functions
Let f=gg+1f=\dfrac{g}{g+1}f=g+1g​, g=2hg=2hg=2h, and h=x+3h=x+3h=x+3. What is f(x)?f(x)?f(x)?

Composition of Functions

Practice: Composition of Functions
Let f=6g2f=6g^2f=6g2, g=h+2g=h+2g=h+2, and h=4x+3h=4x+3h=4x+3. What is f(x)?f(x)?f(x)?

Composition of Functions

Practice: Composition of Functions
Joey is out for dinner with his husband, Carlos. The bill comes and the cost is $45.82\$45.82$45.82 plus tax, which amounts to $51.32\$51.32$51.32. They also told the waiter that they had a loyalty card that allows them 20%20\%20% off the bill including gratuity. 
They are debating whether or not they should tip %15\%15%15 ontop of $45.82\$45.82$45.82 and then use their %20\%20%20 coupon or if they should tip 10%10\%10% ontop of $51.32\$51.32$51.32 and then use their 20%20\%20% off coupon. 

Composition of Functions

Practice: Composition of Functions
Gwen is shopping at Whole Foods. She gets a 10% discount when she uses her loyalty card. She also has a coupon for $35 off her purchase. The store allows you to use them in any order. 
Which is the better deal:

Composition of Functions

Practice: Composition of Functions
Let f(x)=1x+4f(x)=\dfrac{1}{x+4}f(x)=x+41​ and g(x)=1x−2g(x)=\dfrac{1}{x-2}g(x)=x−21​. Determine the following:

Composition of Functions

Practice: Composition of Functions
Let f(x)=x2+1f(x)=x^2+1f(x)=x2+1 and g(x)=xg(x)=\sqrt{x}g(x)=x​. Determine the following:

Combination of Functions

If f(x)=1x−2f(x)=\dfrac{1}{x-2}f(x)=x−21​ and g(x)=23x+4g(x)=\dfrac{2}{3x+4}g(x)=3x+42​, then determine the following:

Combination of Functions

If f(x)=x−2x+1f(x)=\dfrac{x-2}{x+1}f(x)=x+1x−2​ and g(x)=2x+33x−1g(x)=\dfrac{2x+3}{3x-1}g(x)=3x−12x+3​, then determine: