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Questions and Answers
The table below shows the values of a function f(x). What is the value of f(-1) and f^(-1)(-1)?
The table below shows the values of a function f(x). What is the value of f(-1) and f^(-1)(-1)?
- f(-1) = 2, f^(-1)(-1) = 1
- f(-1) = 3, f^(-1)(-1) = 1 (correct)
- f(-1) = 1, f^(-1)(-1) = 3
- f(-1) = 2, f^(-1)(-1) = 2
Which of the following graphs represents a function?
Which of the following graphs represents a function?
- A parabola that opens downwards
- A circle centered at the origin
- A line with a slope of 2 (correct)
- A vertical line
What is the domain and range of the function f(x) = 2x - 1?
What is the domain and range of the function f(x) = 2x - 1?
- Domain: all real numbers, Range: all real numbers (correct)
- Domain: all positive numbers, Range: all real numbers
- Domain: all real numbers, Range: all positive numbers
- Domain: all negative numbers, Range: all positive numbers
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Study Notes
Introduction to Functions
Functions vs. Relations
- A function is a relation in which each input has only one output.
- Not all relations are functions.
Identifying Functions
- A real-world action can be a function, but not all real-world actions are functions.
- Examples of real-world actions that are not functions: throwing a ball, playing a musical instrument.
- Examples of real-world actions that are functions: baking a cake, measuring the length of an object.
Table Representation
- A table can be used to represent a function, where each input corresponds to exactly one output.
- Example: given a table, find the output value for a given input, e.g., find f(x) when x = 2.
- The inverse of a function, denoted by f^(-1), can be found by swapping the input and output values in the table.
Graphical Representation
- A graph can be used to represent a function, where each input corresponds to exactly one output.
- A vertical line test can be used to determine if a graph represents a function: if a vertical line intersects the graph at more than one point, it is not a function.
- Example: identify which graph represents a function.
Domain and Range
- The domain of a function is the set of input values, often represented by x.
- The range of a function is the set of output values, often represented by y.
- Example: find the domain and range of a given graph or function.
Transformations of Functions
- Functions can be transformed by applying vertical or horizontal shifts, reflections, or stretches.
- Example: solve for a transformed function, e.g., f(x) = 2(x + 1) - 3.
Composite Functions
- A composite function is a function that takes another function as its input.
- Example: given two functions, find the composite function, e.g., if f(x) = 2x and g(x) = x + 1, find (f ∘ g)(x).
More Questions
- If f(x) = x^2, what is the value of f(-2)?
- If g(x) = 2x - 1, what is the value of g(3)?
- If h(x) = x^2 + 1, what is the value of h(-1)?
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