Graph Detective
Does the relation y = x^2 pass the Vertical Line Test?
Yes. Every x-value corresponds to only one y-value.
Determine whether f(x) = x^2 + 4 is even, odd, or neither.
Even
Identify the parent function of g(x) = (x - 3)^2 + 5.
Quadratic: f(x) = x^2
If f(x) = x + 3 and g(x) = x^2, find (f + g)(2)
9
What line are the graphs of a function and its inverse reflected across?
y = x
A graph has endpoints at x = -4 and x = 6. What is its domain?
[-4, 6]
Determine whether f(x) = x^3 - 2x is even, odd, or neither.
Odd
Describe the transformation from f(x) = |x| to g(x) = |x + 4| - 2.
Shift left 4 and down 2.
If f(x) = 2x - 1 and g(x) = x + 4, find (f o g)(3).
13
Find the inverse of f(x) = x + 6.
f^(-1)(x) = x - 6
A function decreases on (-infinity, -2), is constant on (-2, 1), and increases on (1, infinity). State the constant interval.
(-2, 1)
Determine whether f(x) = x^2 + x is even, odd, or neither.
Neither
Write an equation for y = x^3 shifted 2 units right and 5 units up.
g(x) = (x - 2)^3 + 5
If f(x) = x + 8 and g(x) = x - 3, find (f o g)(x).
x + 5
Find the inverse of f(x) = 3x - 6.
f^(-1)(x) = (x + 6)/3
For f(x) = -x^2 + 4, describe the end behavior.
As x -> -infinity, f(x) -> -infinity; as x -> infinity, f(x) -> -infinity.
What algebraic test proves that a function is even?
f(-x) = f(x)
Describe the transformations of g(x) = -2sqrt(x - 3) + 1 from y = sqrt(x).
Right 3, vertical stretch by 2, reflect across x-axis, up 1.
If f(x) = 3x and g(x) = x^4, find (g o f)(x).
81x^4
How can you use a graph to determine whether a function has an inverse function?
Use the Horizontal Line Test; every horizontal line must intersect the graph at most once.
DAILY DOUBLE: A graph is increasing on (-3, 2) and decreasing on (2, 7). What important feature occurs at x = 2?
DAILY DOUBLE: A graph is increasing on (-3, 2) and decreasing on (2, 7). What important feature occurs at x = 2?
Determine whether f(x) = x^5 + 3x^3 - x is even, odd, or neither. Justify algebraically.
Odd, because f(-x) = -f(x).
DAILY DOUBLE: Write an equation for y = |x| reflected across the x-axis, vertically stretched by 3, shifted 5 left, and 4 up.
g(x) = -3|x + 5| + 4
Let f(x) = x^2 + 1 and g(x) = 2x - 3. Find (f o g)(-1).
26
Verify whether f(x) = 2x + 5 and g(x) = (x - 5)/2 are inverses.
Yes. f(g(x)) = x and g(f(x)) = x.