Transformations
What is an equation for the squaring function shifted up by 2?
f(x) = x2 + 2
What is an equation for the inverse of the function f(x) = 5x, if the inverse exists?
f(x) = x/5
What are the x-values of the relative extrema of the function f(x) = x3 – 3x2 + x + 1?
relative maximum at x = 0.18; relative minimum at x = 1.8
Using interval notation, state the intervals where the function f(x) = (2x – 3)2 is increasing and where it is decreasing.
decreasing on (–∞,1.5)U(1.5, ∞)
If f(x) = x^2 and g(x) = x – 2, what is the equation for f(g(x))?
f(g(x)) = (x – 2)^2
What is an equation for the absolute value function shifted to the left by 20?
f(x) = |x + 20|
What is the equation for the inverse of the function f(x) = 7x + 1, if the inverse exists?
f(x) = (x – 1)/7
What are the relative and absolute extrema of the function f(x) = x4 + x3 – 2x2 – x + 2?
Absolute minimum at x = –2.4; Relative minimum at x = 0.4; Relative maximum at x = –0.25
Using interval notation, state the intervals where the function f(x) = (2x – 3)2 – x2 is increasing and where it is decreasing.
decreasing on (1.19,1.90); increasing on (–∞,1.19)U(1.90, ∞)
If f(x) = x^2 and g(x) = x – 2, what is the equation for g(f(x))?
g(f(x)) = x^2 – 2
What is an equation for the cubic function stretched vertically by a factor of 4 and reflected across the y-axis?
f(x) = 4(-x)3
What is the equation for the inverse of the function f(x) = x/2 + 15, if the inverse exists?
f(x) = 2(x – 15)
What are the relative extrema of the function f(x) = 0.05x3 – 0.8x3 + 0.4x + 1?
Relative maximum at x = 0.26; relative minimum at x = 10.4
Using interval notation, state the intervals where the function f(x) = 1/(x-5) is increasing and where it is decreasing.
Increasing on (-ထ,5)U(5,ထ)
If f(x) = 2x^2 – x and g(x) = x – 5, what is the equation for f(g(x))?
f(g(x)) = 2(x – 5)^2 – (x – 5)
What is an equation for the quadratic function shifted up by 5, to the right by 4, and reflected across the x-axis?
f(x) = – (x – 4)2 + 5
What is the equation for the inverse of the function f(x) = 2 – 3x2, if the inverse exists?
What are the relative and absolute extrema of the function f(x) = 0.04x4 + 0.2x3 – 1.9x2 + 0.1x + 4?
absolute minimum at x = –7.1; relative maximum at x = 0.026; relative minimum at x = 3.3
Using interval notation, state the intervals where the function f(x) = 0.6x4 + 1.8x3 + x2 – x + 1 is increasing and where it is decreasing.
decreasing on (–∞,1.52)U(–1, 0.27); increasing on (–1.52, –1)U(0.27, ∞)
If f(x) = 2x^2 – x and g(x) = x – 5, what is the equation for g(f(x))?
g(f(x)) = 2x^2 – x – 5
A function, f(x)=-4(x+3)2-5, is shifted up by 3, translated left 2 units, and vertically compressed by a factor of 1/2. What is the equation of the new function?
f(x) = -2(x+5)2-2
What is the test to determine if a relation's inverse is a function?
How do you verify inverses algebraically?
Horizontal Line Test
F and G are inverses if f(g(x))=x and g(f(x))=x
What are the relative and/or absolute extrema of the function f(x) = 0.01(x – 5)4 – 0.1x3 – 0.4x + 4?
Specify if it is a relative or absolute maximum or minimum.
The function has an absolute minimum at x=18.88
Describe increasing and decreasing intervals. Your response must include the direction you read the graph, and an explanation of the change in x as it relates to the change in y.
Must include: Read the graph from left to right
Increasing: As x increases, y increases
Decreasing: As x increases, y decreases
If f(x) = 2x^2 – x, g(x) = x – 5, and h(x) = x/3, what is the equation for h(g(f(x)))?
h(g(f(x))) = (2x2 – x – 5)/3