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Why might simplifying a function not change its domain restrictions?
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Simplifying might eliminate superficial problems, but it doesn't alter inherent domain limitations, like discontinuities or asymptotes.
How are removable discontinuities different from vertical asymptotes?
Removable discontinuities are 'holes' in the graph where the function is undefined, while vertical asymptotes are lines that the graph approaches but never crosses or touches.
How can you determine if a graph represents a function?
By using the vertical line test, where no vertical line should intersect the graph at more than one point.
Explain the concept and significance of the range of a function.
The range is the set of all possible outputs of a function, indicating the extent of values the function can produce.
What does the y = 3x² - 4x + 2 function's graph typically look like?
It is a parabola opening upwards.
Describe an odd function and its symmetry.
An odd function is symmetric about the origin and satisfies the condition f(-x) = -f(x).
What is the importance of identifying domains in functions?
Identifying domains is crucial to avoid undefined expressions such as division by zero or taking square roots of negative numbers.
Give an example of a real-world application of functions.
Finding the volume of a box made from a sheet of cardboard by manipulating its dimensions, demonstrating outputs depending on different inputs.
What kind of graphical test can indicate if outputs are non-unique for different inputs?
The horizontal line test can help determine if different inputs render the same outputs, indicating non-injective behavior.
What is the rule-based way to find the domain of a function with a square root in its equation?
The domain is determined by ensuring that the radicand (the expression inside the square root) is non-negative.
What is the significance of sub-domains in piecewise functions?
They define specific conditions under which different expressions of the function apply, over a specific range of the domain.
What are the domain and range in the context of functions?
The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).
What is a defining characteristic of a function concerning its outputs?
A function must have exactly one output for every input.
What is a piecewise function and give an example?
A piecewise function is defined by multiple sub-functions, each valid over a specific domain. An example is the absolute value function.
What determines if a function is even and what property does its graph exhibit?
A function is even if f(-x) = f(x), and its graph is symmetric about the y-axis.
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