9709 · 1.2
Functions flashcards
Revision flashcards for Cambridge 9709 Functions (syllabus 1.2). Flip, recall, then mark a real past-paper question.
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What is a function?
A rule that maps each element in the domain (input set) to exactly one element in the range (output set).
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What is the 'domain' of a function?
The set of all possible input values (often $x$-values) for which the function is defined.
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What is the 'range' of a function?
The set of all possible output values (often $y$-values or $f(x)$-values) that the function can produce.
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How do you test if a graph represents a function?
Use the vertical line test. If any vertical line intersects the graph more than once, it is not a function.
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What does $fg(x)$ mean?
This is a composite function, also written as $f(g(x))$. You apply the function $g$ first, then apply the function $f$ to the result of $g(x)$.
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What is a one-to-one function?
A function where each output value in the range corresponds to exactly one unique input value in the domain. It passes both the vertical and horizontal line tests.
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When does a function have an inverse, $f^{-1}(x)$?
A function has an inverse if and only if it is a one-to-one function.
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How do you find the inverse of a function $y = f(x)$ algebraically?
1. Swap $x$ and $y$ in the equation. 2. Make the new $y$ the subject of the formula. This new expression for $y$ is $f^{-1}(x)$.
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What is the relationship between the domain of $f(x)$ and the range of $f^{-1}(x)$?
They are identical. The domain of $f(x)$ is the range of $f^{-1}(x)$.
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What is the relationship between the range of $f(x)$ and the domain of $f^{-1}(x)$?
They are identical. The range of $f(x)$ is the domain of $f^{-1}(x)$. This is a crucial and frequently examined point.
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Geometrically, how are the graphs of $y=f(x)$ and $y=f^{-1}(x)$ related?
The graph of $y=f^{-1}(x)$ is a reflection of the graph of $y=f(x)$ in the line $y=x$.
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A common trap: What is the domain of $f(x) = \sqrt{x-5}$?
The expression inside the square root must be non-negative. So, $x-5 \ge 0$, which means the domain is $x \ge 5$.
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For $f(x) = (x-1)^2 + 3$ with domain $x \ge 1$, what is the range?
The vertex is at $(1, 3)$. Since the domain is $x \ge 1$, the function is on the increasing part of the parabola. The minimum value is $f(1)=3$. The range is $f(x) \ge 3$.