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Wize High School Grade 12 Pre-Calculus Textbook > Combination of Functions

Composition of Functions

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Composition of Functions

Functions can be combined through composition.


Let f(x)f(x) and g(x)g(x) be two continuous functions defined on the interval (−∞,∞)(-\infin,\infin).
  • The composition of 2 functions, f(x)f(x) and 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)}.
  • It is read as "ff of gg of xx" or "ff composed of g(x)g(x)".
  • Substitute all values of xx in the function f(x)f(x) with the function 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+1 and g(x)=x+1g(x)=x+1. Find:
  1. f(x)∘g(x)f(x)\circ{}g(x)
  2. g(x)∘f(x)g(x)\circ{}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}

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Example: Composition of Functions

Let f(x)=x−6f(x)=x-6 and g(x)=x2+x+4g(x)=x^2+x+4. Determine:
  1. f(g(x))f(g(x))
  2. g(f(x))g(f(x))
  3. 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}

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}

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}

Practice: Composition of Functions

Let f(x)=1x−2f(x)=\dfrac{1}{x-2} and g(x)=1xg(x)=\dfrac{1}{x}. Determine the following:

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:
  1. Using the 20%20\% discount first then the $10\$10 coupon?
  2. Using the $10\$10 coupon first then the 20%20\% discount?


Practice: Composition of Functions

Let f=3g−1f=3g-1, g=4h+6g=4h+6, and h=9x+10h=9x+10. What is f(x)?f(x)?
Extra Practice
Composition of Functions
Practice: Composition of Functions
Let f=gg+1f=\dfrac{g}{g+1}, g=2hg=2h, and h=x+3h=x+3. What is f(x)?f(x)?
Composition of Functions
Practice: Composition of Functions
Let f=6g2f=6g^2, g=h+2g=h+2, and h=4x+3h=4x+3. What is 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 plus tax, which amounts to $51.32\$51.32. They also told the waiter that they had a loyalty card that allows them 20%20\% off the bill including gratuity.
They are debating whether or not they should tip %15\%15 ontop of $45.82\$45.82 and then use their %20\%20 coupon or if they should tip 10%10\% ontop of $51.32\$51.32 and then use their 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} and g(x)=1x−2g(x)=\dfrac{1}{x-2}. Determine the following:
Composition of Functions
Practice: Composition of Functions
Let f(x)=x2+1f(x)=x^2+1 and g(x)=xg(x)=\sqrt{x}. Determine the following:
Combination of Functions
If f(x)=1x−2f(x)=\dfrac{1}{x-2} and g(x)=23x+4g(x)=\dfrac{2}{3x+4}, then determine the following:
Combination of Functions
If f(x)=x−2x+1f(x)=\dfrac{x-2}{x+1} and g(x)=2x+33x−1g(x)=\dfrac{2x+3}{3x-1}, then determine: