# Function Composition - Video Tutorials & Practice Problems

## Function Composition

Given the functions $f\left(x\right)=\sqrt{x+4}$ and $g\left(x\right)=\left(x-2\right)^2-4$ find $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$

$\left(f\circ g\right)\left(x\right)=\sqrt{x-2}$ ; $\left(g\circ f\right)\left(x\right)=\left(x+4\right)-4\sqrt{x+4}$

$\left(f\circ g\right)\left(x\right)=x-2$ ; $\left(g\circ f\right)\left(x\right)=x\left(x+4\right)$

$\left(f\circ g\right)\left(x\right)=x-2$ ; $\left(g\circ f\right)\left(x\right)=4\sqrt{x-4}$

$\left(f\circ g\right)\left(x\right)=x-2$ ; $\left(g\circ f\right)\left(x\right)=\left(x+4\right)-4\sqrt{x+4}$

Given the functions $f(x)=\frac{1}{x^2-2}$ and $g(x)=\sqrt{x+2}$ find **$(f∘g)(x)$ **and $(g\circ f)(x)$.

$(f\circ g)(x)=\frac{1}{x}$ ; $(g\circ f)(x)=\sqrt{\frac{2x^2-3}{x^2-2}}$

$(f\circ g)(x)=\frac{1}{x}$ ; $(g\circ f)(x)=\sqrt{\frac{3}{x^2-2}}$

$(f∘g)(x)=x$ ; $(g\circ f)(x)=\sqrt{x^2-2}$

$(f∘g)(x)=x$ ; $(g\circ f)(x)=\frac{1}{\sqrt{x+2}-2}$

## Evaluating Composed Functions

Given the functions $f(x)=x+3$ and $g(x)= x^2$ find **$(f∘g)(2)$ **and **$(g∘f)(2)$.**

$(f∘g)(2)=5$ ; $(g∘f)(2)=25$

$(f\circ g)(2)=7;(g\circ f)(2)=4$

$(f∘g)(2)=7$ ; $(g∘f)(2)=25$

$(f∘g)(2)=1$ ; $(g∘f)(2)=1$

## Domain Restrictions of Composed Functions

Given the functions $f(x) = x^2$ and $g(x)=\sqrt{x-8}$ find **$(f∘g)(x)$ **and determine its domain.

$(f∘g)(x)=x-8$ ; $Dom:(-\infty,\infty)$

$(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:(-∞,∞)$

$(f∘g)(x)=x-8$ ; $Dom:[8,∞)$

$(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:[8,∞)$

## Decomposition of Functions

## Decomposition of Functions Example 1

## Do you want more practice?

- In Exercises 1–30, find the domain of each function. f(x)=-2(x+5)
- In Exercises 1–30, find the domain of each function. g(x) = 3/(x-4)
- Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2 ...
- In Exercises 1–30, find the domain of each function. g(x) = 2/(x+5)
- Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2 ...
- Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2 ...
- Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ+g)(3)
- In Exercises 1–30, find the domain of each function. f(x) = 1/(x^2+1) - 1/(x^2-1)
- In Exercises 1–30, find the domain of each function. h(x) = 4/(3/x - 1)
- Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒg)(-3)
- Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ/g)(-1)
- For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-8
- For the pair of functions defined, find (ƒg)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-7
- For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x...
- For the pair of functions defined, find (ƒ-g)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x...
- In Exercises 1–30, find the domain of each function. f(x) = √(5x+35)
- For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2. ƒ(x)=√(4x-1), g(x)=1/...
- In Exercises 1–30, find the domain of each function. f(x) = √(24 - 2x)
- For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2. ƒ(x)=√(4x-1), g(x)=1/...
- In Exercises 1–30, find the domain of each function. f(x) = (2x+7)/(x^3 - 5x^2 - 4x+20)
- In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1
- In Exercises 31–50, find fg and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1
- In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1
- In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = x -5, g(x) = 3x²
- Use the graph to evaluate each expression. See Example 3(a). (ƒ/g)(1)
- Use the graph to evaluate each expression. See Example 3(a). (ƒg)(0)
- Use the graph to evaluate each expression. See Example 3(a). (ƒ+g)(0)
- Use the graph to evaluate each expression. See Example 3(a). (ƒ/g)(2)
- In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1
- In Exercises 31–50, find fg and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1
- In Exercises 31–50, find f−g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1
- In Exercises 31–50, find f/g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 18
- In Exercises 31–50, find fg and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 17
- In Exercises 31–50, find fg and determine the domain for each function. f(x) = √x, g(x) = x − 4
- In Exercises 31–50, find f−g and determine the domain for each function. f(x) = √x, g(x) = x − 4
- In Exercises 31–50, find fg and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x
- In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x
- In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -...
- For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=2-x
- In Exercises 31–50, find f/g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -...
- In Exercises 31–50, find fg and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)
- In Exercises 31–50, find f−g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)
- In Exercises 31–50, find f/g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)
- In Exercises 31–50, find fg and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)
- For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=-2x+5
- In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) =...
- In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) =...
- For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1/x
- In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) =...
- In Exercises 51–66, find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2). f(x) = 2x, g(x) = x+7
- In Exercises 51–66, find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2). f(x) = x+4, g(x) = 2x + 1
- For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=1-x^2
- For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=x^2+3x+1
- In Exercises 51–66, find a. (fog) (2) b. (go f) (2) f(x)=4x-3, g(x) = 5x² - 2
- In Exercises 51–66, find a. (fog) (2) b. (go f) (2) f(x) = x²+2, g(x) = x² – 2
- Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘g)(-2)
- In Exercises 51–66, find a. (fog) (2) b. (go f) (2) f(x) = 4-x, g(x) = 2x² +x+5
- In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5
- Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (g∘ƒ)(0)
- In Exercises 51–66, find c. (fog) (2) d. (go f) (2). f(x) = √x, g(x) = x − 1
- Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘ƒ)(2)
- In Exercises 51–66, find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2). f(x) = 2x-3, g(x) = (x+3)/2
- In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function withou...
- In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function withou...
- In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = √x, g(x) = x − 2
- In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = x² + 4, g(x) = √(1 − x)
- Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=8x+12, g(x)=3x-1
- In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog...
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=x^3, g(x)=x^2+3x-1
- Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x^3, g(x)=x^2+3x-1
- In Exercises 76–81, find the domain of each function. g(x) = 4/(x - 7)
- Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4
- Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√(x-1), g(x)=3x
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=√(x-1), g(x)=3x
- In Exercises 76–81, find the domain of each function. f(x) = x/(x^2 + 4x -21)
- In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog...
- Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1
- In Exercises 82–84, find f + g, f - g, fg, and f/g. f(x) = x^2 + x + 1, g(x) = x^2 -1
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7...
- Use the graphs of f and g to solve Exercises 83–90. Find (f+g)(−3).
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7...
- Use the graphs of f and g to solve Exercises 83–90. Find (g-f) (-2).
- Given functions f and g, (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=1/(x-2), g(x)=1/x
- Use the graphs of f and g to solve Exercises 83–90. Find the domain of ƒ + g.
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=1/(x-2), g(x)=1/x
- Use the graphs of f and g to solve Exercises 83–90. Find the domain of ƒ/g.
- In Exercises 91–94, use the graphs of f and g to evaluate each composite function. (go f) (0)
- In Exercises 95–96, find all values of x satisfying the given conditions. f(x) = 2x − 5, g(x) = x² − 3x + 8, a...
- If (fg)(x) = 4x²−x−5, find f and g.
- Let ƒ(x) = 3x^2 - 4 and g(x) = x^2 - 3x -4. Find each of the following. (f/g)(-1)
- Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (g ○ ƒ)(x)
- Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (g ○ ƒ)(3)
- Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. the domain of ƒ ○ g
- Use the table to evaluate each expression, if possible. (ƒ+g)(1)
- Use the table to evaluate each expression, if possible. (f-g)(3)
- Use the tables for ƒ and g to evaluate each expression. (g∘ƒ)(-2)
- Use the tables for ƒ and g to evaluate each expression. (ƒ∘g)(3)