# Intro to Functions & Their Graphs - Video Tutorials & Practice Problems

## Relations and Functions

State the inputs and outputs of the following relation. Is it a function? {$\left(-3,5\right),\left(0,2\right),\left(3,5\right)$}

State the inputs and outputs of the following relation. Is it a function? {$\left(2,5\right),\left(0,2\right),\left(2,9\right)$}

## Relations & Functions Example 1

## Verifying if Equations are Functions

Is the equation $y=-2x+10$ a function? If so, rewrite it in function notation and evaluate at $f\left(3\right)$.

$f\left(3\right)=4$ , **Is A Function**

$f\left(3\right)=3$, **Is A Function**

$f\left(3\right)=1$, **Is A Function**

**Is NOT A Function**

Is the equation $y^2+2x=10$ a function? If so, rewrite it in function notation and evaluate at $f\left(-1\right)$.

$f\left(-1\right)=\sqrt{12}$, **Is A Function**

$f\left(-1\right)=12$, **Is A Function**

$f\left(-1\right)=\frac92$, **Is A Function**

**Is NOT A Function**

## Finding the Domain and Range of a Graph

Find the domain and range of the following graph (write your answer using interval notation).

**Dom: $\left[-5,5\right]$ , Ran: **$\left[-4.4\right]$

**Dom: $\left[-2,2\right]$ , Ran: **$\left[-3,3\right]$

**Dom: $\left[-4,4\right]$ , Ran: **$\left[-5,5\right]$

**Dom: $\left(-5,5\right)$ , Ran: **$\left(-4,4\right)$

## Finding the Domain of an Equation

Find the domain of $f\left(x\right)=\sqrt{x+4}$ . Express your answer using interval notation.

**Dom: **$\left[4,\infty\right)$

**Dom: **$\left[-4,2\right]$

**Dom: **$\left[2,4\right]$

**Dom: **$\left[-4,\infty\right)$

Find the domain of $f\left(x\right)=\frac{1}{x^2-5x+6}$ . Express your answer using interval notation.

**Dom: **$\left(-\infty,2\right)\cup\left(2,\infty\right)$

**Dom: **$\left(-\infty,\infty\right)$

**Dom: **$\left(-2,2\right)\cup\left(2,3\right)\cup\left(3,\infty\right)$

**Dom: **$\left(-\infty,2\right)\cup\left(2,3\right)\cup\left(3,\infty\right)$

## Do you want more practice?

- In Exercises 1–30, find the domain of each function. f(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 ...
- 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) = 3/(x^2-2x-15)
- In Exercises 1–30, find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)
- Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ+g)(-5)
- Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ-g)(4)
- In Exercises 1–30, find the domain of each function. f(x) = √(x - 3)
- Let ƒ(x)=x^2+3 and g(x)=-2x+6. Find each of the following. See Example 1. (ƒ/g)(5)
- In Exercises 1–30, find the domain of each function. f(x) = 1/√(x - 3)
- For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2. ƒ(x)=3x+4, g(x)=2x-6
- 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 (ƒg)(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)=√(4x-1), g(x)=1/...
- In Exercises 1–30, find the domain of each function. g(x) = √(x −2)/(x-5)
- 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 f/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)
- In Exercises 31–50, find fg 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). (ƒg)(1)
- 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 f−g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 16
- In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15
- In Exercises 31–50, find ƒ+g 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 ƒ-g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x
- In Exercises 31–50, find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2...
- In Exercises 31–50, find f−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)=6x+2
- 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 ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)
- 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^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
- 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
- Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (ƒ∘g)(4)
- In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = x²+2, g(x) = x² – 2
- In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5
- 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 51–66, find a. (fog) (x) b. (go f) (x). 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 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) = 2/(x+3), g(x) = 1/x
- In Exercises 67-74, find a. (fog) (x) b. the domain of f o g. f(x) = x/(x+1), g(x) = 4/x
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7
- Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=8x+12, g(x)=3x-1
- Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=x+3
- In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog...
- In Exercises 76–81, find the domain of each function. g(x) = 4/(x - 7)
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4
- 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
- Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1
- Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=1/(x+5)
- Use the graphs of f and g to solve Exercises 83–90. Find (fg) (2).
- Use the graphs of f and g to solve Exercises 83–90. Find(g/f)(3)
- In Exercises 89–90, express the given function h as a composition of two functions f and g so that h(x) = (f ○...
- Use the graphs of f and g to solve Exercises 83–90. Graph f-g.
- In Exercises 91–94, use the graphs of f and g to evaluate each composite function. (fog) (-1)
- Let ƒ(x) = 3x^2 - 4 and g(x) = x^2 - 3x -4. Find each of the following. (f+g)(2k)
- Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (ƒ ○ g)(x)
- Let ƒ(x) = √(x-2) and g(x) = x^2. Find each of the following, if possible. (f ○ g)(-6)
- Use the table to evaluate each expression, if possible. (f-g)(3)
- The graphs of two functions ƒ and g are shown in the figures. Find (g∘ƒ)(3).