Functions and Graphs Flashcards
What is a function?
A function is a rule that assigns to each input exactly one output.
What is the domain of a function?
The set of all valid input values (x-values).
What is the range of a function?
The set of all possible output values (y-values).
What is a one-to-one function?
A function where each output is mapped from a unique input.
What is a many-to-one function?
A function where different inputs can give the same output.
What is the notation f : x ↦ x²?
This means f is a function that maps x to x².
What is the inverse of a function f(x)?
A function that reverses f(x), written f⁻¹(x).
What condition must a function satisfy to have an inverse?
It must be one-to-one.
How do you find the inverse of a function?
Swap x and y, then solve for y.
How is the graph of y = f⁻¹(x) related to y = f(x)?
It is a reflection of y = f(x) in the line y = x.
What does the composition (f ∘ g)(x) mean?
It means f(g(x)); apply g first, then f.
What does the graph of a modulus function y = |f(x)| look like?
Any part of the graph below the x-axis is reflected above it.
What is the transformation y = |x|?
A V-shaped graph with vertex at the origin.
How do you represent functions graphically?
By plotting the output value (y) for each x-value in the domain.
How do you solve f(x) = g(x) graphically?
Plot both functions and find their points of intersection.
What does y = f(x) + a represent?
A vertical shift of a units up.
What does y = f(x + a) represent?
A horizontal shift of a units to the left.
What is a step function?
A function that jumps between constant values (e.g., greatest integer function).
What is a piecewise function?
A function defined by different expressions over different intervals.
What is a composite function?
A function made by applying one function to the result of another.
