MTH130 1.2 Functions and Graphs
A. Relations, Functions, Domain and Range
Relation: any set of ordered pairs (x, y) is called a relation in x and y. Some relations are functions and some are not
The domain is all the x values
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The range is all the y values
o Hint: “d” comes before “r” in the alphabet just like x comes before y so “domain” goes with x and “range” goes with y.
o Sometimes we can make a list of all the x and y values and use this notation to identify the domain and range:
D = { , , , } and R = { , , , }
o Sometimes we can’t make a list of all the x and y values and we use interval notation.
For the domain, both numbers in the interval are from the x-axis.
For the range, both numbers in the interval are from the y-axis.
We use parenthesis ( , ) for the interval when the small or large numbers cannot be included. On a graph we see this as an open circle.
We use brackets [ , ] for the interval when the small or large numbers are included in the domain and range. On a graph we see this as a closed circle.
If the small number for the interval is -∞ or the large number for the interval is ∞ we always use ( or ) for the interval.
Function: a rule that assigns to each element, x, in a set A, exactly one element, f(x), in a set B.
This means a function CANNOT repeat any x-coordinates but CAN repeat y-coordinates.
The Vertical Line Test can be used to determine if a graphed relation is a function. If a vertical line passes through the graph two or more times, the graph fails the vertical line test which means the relation is NOT a function.
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Determine if the relation is a function. Then find the domain and range.
1. {(1, 2), (3, 4), (5, 6), (7, 8)}
2. {(2, n), (5, p), (1, 2), (5, m)}
3. {(3, 2), (-1, 2), (-5, 4), (7, 8)}
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Use the Vertical Line Test to determine if the relation is a function. 9. 10.
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Write the domain and range using interval notation. 13. 14.
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B. Evaluating Functions
19. If 𝑓(𝑥)=−𝑥2 +5 find𝑓(−4)
20. If 𝑔(𝑥)=−3𝑥−4 find𝑔(2)
21. If h(𝑥) = 2𝑥3 − 1 find h(−3)
22. If 𝑘(𝑥) = 𝑥−4 find 𝑘(−2) 𝑥2−5𝑥−36
23. If 𝑓(𝑥)=3𝑥2 −5 find𝑓(4𝑛)
24. If h(𝑥)= 𝑥+3 findh(𝑥+1) 2𝑥+1
25. If 𝑓(𝑥)=3𝑥2 −2𝑥 find𝑓(−𝑥)
26. If 𝑔(𝑥)=𝑥−3 find𝑔(𝑧+5)
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C. Finding the domain of an equation
1) Is it a fraction?
a. Find the excluded values. (Set the denominator
equal to zero and solve for x. )
b. Make a list of all the excluded values.
2) Is it an even index radical?
a. Set the inside expression 0 and solve for x. b. Domain: ___, oritcouldbe,___
Find all the values of x that are NOT in the domain. If there is more than one value, separate them with commas.
27. 𝑓(𝑥) = 2𝑥+1 28. 𝑔(𝑥) = 𝑥−5
(𝑥−9)(𝑥+7)
Find the domain of the function. Write your answer using interval notation. 29. 𝑓(𝑥)=√𝑥+4 30. 𝑔(𝑥)=√𝑥−3
31. 𝑓(𝑥)=√3−4𝑥 32. 𝑔(𝑥)=√−𝑥−8
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D. Finding inputs and outputs of a function from its graph.
33. The graph of a function, 𝑓(𝑥) is given. Find all information.
a. Find 𝑓(−1)
b. Find 𝑓(2)
c. Find 𝑓(6)
d. For what value of x is 𝑓(𝑥) = 3 ? e. For what value(s) of x is 𝑓(𝑥) = 0 ?
34. The graph of a function, h(𝑥) is given. Find all information. a. Find h(0)
b. Find h(−2)
c. Find h(5)
d. For what value(s) of x is h(𝑥) = 0 ? e. For what value of x is h(𝑥) = −4 ?
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