Functions - ACT Math
Card 1 of 60
What transformation does $f(x) + k$ represent?
What transformation does $f(x) + k$ represent?
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Vertical shift: up $k$ units if $k > 0$, down $|k|$ units if $k < 0$
Vertical shift: up $k$ units if $k > 0$, down $|k|$ units if $k < 0$
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If $f(x) = 2x + 4$, find $f^{-1}(x)$.
If $f(x) = 2x + 4$, find $f^{-1}(x)$.
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$f^{-1}(x) = \dfrac{x - 4}{2}$ (swap $x$ and $y$, solve for $y$: $x = 2y + 4 \Rightarrow y = \dfrac{x-4}{2}$)
$f^{-1}(x) = \dfrac{x - 4}{2}$ (swap $x$ and $y$, solve for $y$: $x = 2y + 4 \Rightarrow y = \dfrac{x-4}{2}$)
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What is the vertex of $f(x) = (x - 3)^2 + 2$?
What is the vertex of $f(x) = (x - 3)^2 + 2$?
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$(3, 2)$
$(3, 2)$
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What is the domain of $f(x) = \dfrac{1}{x}$?
What is the domain of $f(x) = \dfrac{1}{x}$?
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All real numbers except $x = 0$ (or $x \neq 0$)
All real numbers except $x = 0$ (or $x \neq 0$)
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If $f(x) = x^2$, describe the graph of $g(x) = -x^2$.
If $f(x) = x^2$, describe the graph of $g(x) = -x^2$.
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Same as $f(x)$ but reflected across the $x$-axis (opens downward)
Same as $f(x)$ but reflected across the $x$-axis (opens downward)
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What is the range of a function?
What is the range of a function?
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The set of all possible outputs $f(x)$ produced by inputs from the domain.
The set of all possible outputs $f(x)$ produced by inputs from the domain.
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If $f(x) = 4x - 1$, find $f(0)$.
If $f(x) = 4x - 1$, find $f(0)$.
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$f(0) = 4(0) - 1 = -1$
$f(0) = 4(0) - 1 = -1$
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What does the notation $f(x)$ mean?
What does the notation $f(x)$ mean?
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A function named $f$ with input $x$; read as "$f$ of $x$"
A function named $f$ with input $x$; read as "$f$ of $x$"
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What is the $y$-intercept of $f(x) = -2x + 5$?
What is the $y$-intercept of $f(x) = -2x + 5$?
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$b = 5$ (point $(0, 5)$)
$b = 5$ (point $(0, 5)$)
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If $f(x) = x^2$ and $g(x) = x - 1$, find $f(g(2))$.
If $f(x) = x^2$ and $g(x) = x - 1$, find $f(g(2))$.
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$g(2) = 2 - 1 = 1$, then $f(1) = 1^2 = 1$
$g(2) = 2 - 1 = 1$, then $f(1) = 1^2 = 1$
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What is the vertex form of a quadratic function?
What is the vertex form of a quadratic function?
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$f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex
$f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex
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What transformation does $af(x)$ represent when $a > 1$?
What transformation does $af(x)$ represent when $a > 1$?
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Vertical stretch by factor $a$
Vertical stretch by factor $a$
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If $g(x) = x^2 - 4$, find $g(3)$.
If $g(x) = x^2 - 4$, find $g(3)$.
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$g(3) = 3^2 - 4 = 9 - 4 = 5$
$g(3) = 3^2 - 4 = 9 - 4 = 5$
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If $f(x) = x^2$, describe the graph of $g(x) = 3x^2$.
If $f(x) = x^2$, describe the graph of $g(x) = 3x^2$.
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Same as $f(x)$ but vertically stretched by factor of 3 (narrower parabola)
Same as $f(x)$ but vertically stretched by factor of 3 (narrower parabola)
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What is the domain of $f(x) = \sqrt{x}$?
What is the domain of $f(x) = \sqrt{x}$?
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$x \geq 0$ (in the real numbers)
$x \geq 0$ (in the real numbers)
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What is an inverse function $f^{-1}(x)$?
What is an inverse function $f^{-1}(x)$?
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A function that "undoes" $f(x)$; if $f(a) = b$, then $f^{-1}(b) = a$. An inverse $f^{-1}$ exists on a domain where $f$ is one-to-one (passes the horizontal line test). Inverses swap domain/range and reflect across $y=x$.
A function that "undoes" $f(x)$; if $f(a) = b$, then $f^{-1}(b) = a$. An inverse $f^{-1}$ exists on a domain where $f$ is one-to-one (passes the horizontal line test). Inverses swap domain/range and reflect across $y=x$.
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The graph of $f(x)$ passes through $(2, 5)$. What point does the graph of $f(x) + 3$ pass through?
The graph of $f(x)$ passes through $(2, 5)$. What point does the graph of $f(x) + 3$ pass through?
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$(2, 8)$ (same $x$, $y$ increases by 3)
$(2, 8)$ (same $x$, $y$ increases by 3)
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If $h(x) = \dfrac{x}{2} + 1$, find $h(8)$.
If $h(x) = \dfrac{x}{2} + 1$, find $h(8)$.
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$h(8) = \dfrac{8}{2} + 1 = 4 + 1 = 5$
$h(8) = \dfrac{8}{2} + 1 = 4 + 1 = 5$
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Does the graph of $y = 2x + 1$ represent a function?
Does the graph of $y = 2x + 1$ represent a function?
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Yes (passes vertical line test; each $x$ has exactly one $y$)
Yes (passes vertical line test; each $x$ has exactly one $y$)
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What is the axis of symmetry for a parabola?
What is the axis of symmetry for a parabola?
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The vertical line that passes through the vertex; for $f(x) = a(x - h)^2 + k$, it's $x = h$
The vertical line that passes through the vertex; for $f(x) = a(x - h)^2 + k$, it's $x = h$
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