Functions and inverses, Domains and Ranges
How can you check the graph of an equation to determine whether it is a function or not?
Functions cannot have more than one output (y-value) for each unique input (x-value).
This means that if a vertical line passes through the graph anyway, there are duplicate y-values at the same value of x, meaning it is not a function.
f(x) = 2x + 5
g(x) = -x + 0.6
What is (f + g)(x) equal to?
(f + g)(x) = x + 5.6
If you shift the graph of the function y = x2 + 9x + 5 up two units, what would the new function be?
y = x2 + 9x + 7
Is y = 2x an odd function, even function, or neither?
odd function, it is symmetric about the origin. For example, if you plugin x = 3, then y = 6
However, if you plugin x = -3, then y = -6
What middle school did I go to?
a) Bel Air
b) Fallston
c) C Milton
d) Patterson Mill
d) Patterson Mill
What is the domain and range of any line that is not horizontal or vertical?
Both all real numbers, because a line will go up and down forever, and left and right forever.
f(x) = 2x + 9
g(x) = 5x - 10
What is (f - g)(x) equal to?
Because 2x + 9 - (5x - 10) = 2x + 9 - 5x + 10 = -3x + 19
If you shift the graph of the function y = x2 - x - 3 down two units, what would the new function be?
y = x2 - x - 5
Is y = x3 an odd function, even function, or neither?
Odd function, it is symmetric about the origin. For example, if you plugin x = 2, then y = 8
However, if you plugin x = -2, then y = -8
What color is my car?
a) white
b) a shade of blue
c) a shade of red
d) silver/grey
c) a shade of red
What is the domain and range of the function y = x2
Graph it on desmos or your calculator if needed.
Domain: All real numbers
Range: y ≥ 0, because you can never square a number and get a negative.
f(x) = x + 7
g(x) = -2x + 5
What is (g·f)(x) equal to?
(g·f)(x) = (x + 7)(-2x + 5) = -2x2 + 5x - 14x + 35
= -2x2 - 9x + 35
If you shift the graph of the function y = 4x2 + 3x - 6 right two units, what would the new function be?
y = 4(x - 2)2 + 3(x - 2) - 6
Or if you simplify, 4(x2 - 4x + 4) + 3x - 6 - 6 =
4x2 - 13x + 4
Is y = x2 - 4 an odd function, even function, or neither?
Even. It is symmetric across the x-axis, meaning if you plug in -x you get the same answer for y as if you plugged in x.
What day was my birthday?
a) February 20th
b) February 25th
c) February 15th
d) February 18th
a) February 20th
Are the following two functions inverses?
f(x) = -0.5x - 6
g(x) = -2x - 12
Yes, f-1(x) = -2x - 3 because...
x = -0.5y - 6 --> -0.5y = x + 6 --> y = -2x - 12
f(x) = x2 - 3x - 40
g(x) = x - 8
Find (f/g)(x) and simplify by factoring! Then, find (f/g)(2)
(f/g)(x) =(x - 8)(x + 5)/(x - 8) = x + 5
So (f/g)(2) = 2 + 5 = 7
If you shift the graph of the function y = -x2 - 7x + 10 left two units, what would the new function be?
y = -(x + 2)2 - 7(x + 2) + 10
Or if you simplify, y = -(x2 + 4x + 4) - 7x - 14 + 10 = -x2 - 11x -8
Is y = x - 5 an odd function, even function, or neither?
Neither. This is not symmetric about the origin or y-axis. For example, if you plugin x = 6, then y = 1
If you plugin x = -6, then y = -11. (y should equal 1 if it is even, and -1 if it is odd)
Name the top 3 largest states by land area in the United States, in order of largest to smallest.
Must be in the correct order to earn points.
Alaska (665,000 mi2), Texas (269,000 mi2), then California (164,000 mi2)
Next largest are Montana (147,000 mi2), New Mexico (122,000 mi2), and Arizona (114,000 mi2)
Are the following two functions inverses?
f(x) = 2x + 5
g(x) = x - 2.5
No, f-1(x) = 0.5x - 2.5 because...
x = 2y + 5 --> 2y = x - 5 --> y = 0.5x - 2.5
f(x) = x - 3
g(x) = 4x + 1
Find the composite function g(f(x)), then find g(f(-1))
g(f(x)) = 4(x - 3) + 1 = 4x - 12 + 1 = 4x - 11
So g(f(-1)) = 4(-1) - 11 = -4 - 11 = -15
If you shift the graph of the function y = 7x + 10 left four units and up five units, what would the new function be?
y = 7(x + 4) + 15 OR if you simplify, 7x + 28 + 15 = 7x + 43
Given the equation y = 3x2 + 5x - 11, what would be the equation if we reflected its graph across the x-axis, and what would be the equation reflected across the y-axis?
x-axis reflection: y = -(3x2 + 5x - 11) OR y = -3x2 - 5x + 11
y-axis reflection: y = 3(-x)2 + 5(-x) - 11 OR y = 3x2 - 5x - 11
What sport have I not played, either in school or rec?
a) Soccer
b) Tennis
c) Golf
d) Baseball
b) Tennis