Function Basics
What is a function?
A relation where each input has exactly one output.
What is this equation telling you: y = (x - 4 )^2?
Shift graph 4 points to the right.
How does this parent function look like: y = x^2?
U shape
Is this a function or not: {(1,2),(2,3),(3,4),(4,5)}?
Yes it is a function.
A student says:
"The function f(x) = 1/x-3 has domain all real numbers."
Error: They forgot that X does not equal to 3 because the denominator cannot be 0.
What is the domain?
The set of all possible x-values of a function.
What is this equation telling you: y = 1/3 ( x + 5)^3 - 2?
Shift left 5 points, move 2 points down, Vertical shrink by 1/3
How does this parent function look like: y = cube root?
A sideways S, Passes through (0,0), goes into both left and right sides.
Is this a function or not: {(0,7),(2,7),(2,9),(4,11)}?
No it is not a function.
A student writes:
"If f(x) = 2x + 5, then f(2x) = (2) f(x)"
Error: Misapplied function notation. f(2x) = 2(2x) + 5 = 4x +5, not 2f(x).
What is the vertical line test?
A quick test using vertical lines to see if a graph is a function.
What is this equation telling you: y = |x - 3| + 8?
Shift right 3 points, Move up 8 points.
Two curved branches, One in the top right, One in the bottom-left, never touches the axes.
Is this a function or not: Vertical line at x = 6?
No it is not function.
A student says:
"The range of f(x) = X^2 is all real numbers."
Error: The range is only y greater than or equal to 0, not all real numbers.
What is a output?
The value of a function at a certain x-value.
What is this equation telling you: y = 2cos(2x)?
Vertical stretch by 2, Horizontal shrink by factor of 2.
How does this parent function look like: y = cos x?
Wave, starts at the top, same shape as sine just shifted left.
Is this a function or not:
x: -2, -1, 0, 1
y: 5, 3, 1, -1
A student claims:
"The inverse of f(x) = x^2 is f^-1(x) = square root of x"
Error: f(x) = x^2 is not one-to-one over all real numbers; inverse must be restricted to x greater than or equal to 0 (or x less than or equal to 0).
What is a one-to-one function?
A function where each element of the range is paired with exactly one element of the domain and the function passes both the vertical and horizontal line tests.
What is this equation telling you: y = -sin ( x - pi/2)?
Shift pi/2 points to the right, Reflection over x-axis
How does this parent function look like: y = sin x?
A smooth wave, goes up, then down, repeating, max height = 1, min = -1.
Is this is function or not:
x: 1, 2, 3, 1
y: 4, 8, 12, 16
No this is not a function.
A student says:
"The graph of y = (x - 2)^2 + 3 is shifted 2 units up and 3 units right."
Error: Horizontal shift and vertical shift are reversed. Correct: 2 right, 3 up.