Composition of Functions - AP Precalculus
Card 1 of 30
If $f(x) = 2x$ and $g(x) = x + 3$, what is $(f \bigcirc g)(x)$?
If $f(x) = 2x$ and $g(x) = x + 3$, what is $(f \bigcirc g)(x)$?
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$(f \bigcirc g)(x) = 2(x + 3)$. Substitute $g(x) = x + 3$ into $f$.
$(f \bigcirc g)(x) = 2(x + 3)$. Substitute $g(x) = x + 3$ into $f$.
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If $f(x) = x^2$ and $g(x) = 3x + 1$, find $(g \bigcirc f)(x)$.
If $f(x) = x^2$ and $g(x) = 3x + 1$, find $(g \bigcirc f)(x)$.
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$(g \bigcirc f)(x) = 3x^2 + 1$. Apply $g$ to $f(x) = x^2$: $g(x^2) = 3x^2 + 1$.
$(g \bigcirc f)(x) = 3x^2 + 1$. Apply $g$ to $f(x) = x^2$: $g(x^2) = 3x^2 + 1$.
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What is $(f \bigcirc g)(2)$ if $f(x) = 3x$ and $g(x) = x - 1$?
What is $(f \bigcirc g)(2)$ if $f(x) = 3x$ and $g(x) = x - 1$?
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$(f \bigcirc g)(2) = 3$. $g(2) = 1$, then $f(1) = 3$.
$(f \bigcirc g)(2) = 3$. $g(2) = 1$, then $f(1) = 3$.
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Find $(g \bigcirc f)(0)$ for $f(x) = x^2 + 1$ and $g(x) = 2x$.
Find $(g \bigcirc f)(0)$ for $f(x) = x^2 + 1$ and $g(x) = 2x$.
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$(g \bigcirc f)(0) = 2$. $f(0) = 1$, then $g(1) = 2$.
$(g \bigcirc f)(0) = 2$. $f(0) = 1$, then $g(1) = 2$.
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Calculate $(f \bigcirc g)(x)$ if $f(x) = 2x + 1$ and $g(x) = x^3$.
Calculate $(f \bigcirc g)(x)$ if $f(x) = 2x + 1$ and $g(x) = x^3$.
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$(f \bigcirc g)(x) = 2x^3 + 1$. Apply $f$ to $g(x) = x^3$: $2x^3 + 1$.
$(f \bigcirc g)(x) = 2x^3 + 1$. Apply $f$ to $g(x) = x^3$: $2x^3 + 1$.
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If $f(x) = x^2 + 2$ and $g(x) = 4x$, find $(f \bigcirc g)(x)$.
If $f(x) = x^2 + 2$ and $g(x) = 4x$, find $(f \bigcirc g)(x)$.
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$(f \bigcirc g)(x) = 16x^2 + 2$. Substitute $g(x) = 4x$ into $f$: $(4x)^2 + 2$.
$(f \bigcirc g)(x) = 16x^2 + 2$. Substitute $g(x) = 4x$ into $f$: $(4x)^2 + 2$.
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If $f(x) = 2x + 1$ and $g(x) = x - 3$, find $(f \bigcirc g)(x)$.
If $f(x) = 2x + 1$ and $g(x) = x - 3$, find $(f \bigcirc g)(x)$.
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$(f \bigcirc g)(x) = 2(x - 3) + 1$. Substitute $g(x) = x - 3$ into $f$.
$(f \bigcirc g)(x) = 2(x - 3) + 1$. Substitute $g(x) = x - 3$ into $f$.
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Determine $(g \bigcirc f)(x)$ if $f(x) = x + 1$ and $g(x) = x^3$.
Determine $(g \bigcirc f)(x)$ if $f(x) = x + 1$ and $g(x) = x^3$.
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$(g \bigcirc f)(x) = (x + 1)^3$. Apply $g$ to $f(x) = x + 1$: $(x + 1)^3$.
$(g \bigcirc f)(x) = (x + 1)^3$. Apply $g$ to $f(x) = x + 1$: $(x + 1)^3$.
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Find $(g \bigcirc f)(x)$ if $f(x) = x + 2$ and $g(x) = 3x$.
Find $(g \bigcirc f)(x)$ if $f(x) = x + 2$ and $g(x) = 3x$.
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$(g \bigcirc f)(x) = 3(x + 2)$. Apply $g$ to $f(x) = x + 2$: $3(x + 2)$.
$(g \bigcirc f)(x) = 3(x + 2)$. Apply $g$ to $f(x) = x + 2$: $3(x + 2)$.
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If $f(x) = x - 4$ and $g(x) = x^2$, find $(f \bigcirc g)(x)$.
If $f(x) = x - 4$ and $g(x) = x^2$, find $(f \bigcirc g)(x)$.
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$(f \bigcirc g)(x) = x^2 - 4$. Apply $f$ to $g(x) = x^2$: $x^2 - 4$.
$(f \bigcirc g)(x) = x^2 - 4$. Apply $f$ to $g(x) = x^2$: $x^2 - 4$.
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Write $(f \bigcirc g)(x)$ for $f(x) = x^3$ and $g(x) = \frac{1}{x}$.
Write $(f \bigcirc g)(x)$ for $f(x) = x^3$ and $g(x) = \frac{1}{x}$.
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$(f \bigcirc g)(x) = (1/x)^3$. Substitute $g(x) = \frac{1}{x}$ into $f(x) = x^3$.
$(f \bigcirc g)(x) = (1/x)^3$. Substitute $g(x) = \frac{1}{x}$ into $f(x) = x^3$.
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Determine $(f \bigcirc g)(x)$ if $f(x) = 1/x$ and $g(x) = x - 5$.
Determine $(f \bigcirc g)(x)$ if $f(x) = 1/x$ and $g(x) = x - 5$.
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$(f \bigcirc g)(x) = 1/(x - 5)$. Apply $f$ to the result $g(x) = x - 5$.
$(f \bigcirc g)(x) = 1/(x - 5)$. Apply $f$ to the result $g(x) = x - 5$.
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If $f(x) = 2x + 3$, find $(f \bigcirc f)(x)$.
If $f(x) = 2x + 3$, find $(f \bigcirc f)(x)$.
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$(f \bigcirc f)(x) = 2(2x + 3) + 3$. Substitute $f(x)$ into itself.
$(f \bigcirc f)(x) = 2(2x + 3) + 3$. Substitute $f(x)$ into itself.
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Find $(g \bigcirc f)(x)$ if $f(x) = x^2$ and $g(x) = \frac{1}{x}$.
Find $(g \bigcirc f)(x)$ if $f(x) = x^2$ and $g(x) = \frac{1}{x}$.
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$(g \bigcirc f)(x) = \frac{1}{x^2}$. Apply $g$ to $f(x) = x^2$: $\frac{1}{x^2}$.
$(g \bigcirc f)(x) = \frac{1}{x^2}$. Apply $g$ to $f(x) = x^2$: $\frac{1}{x^2}$.
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If $f(x) = x + 1$ and $g(x) = x^2$, what is $(f \bigcirc g)(-2)$?
If $f(x) = x + 1$ and $g(x) = x^2$, what is $(f \bigcirc g)(-2)$?
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$(f \bigcirc g)(-2) = 5$. $g(-2) = 4$, then $f(4) = 5$.
$(f \bigcirc g)(-2) = 5$. $g(-2) = 4$, then $f(4) = 5$.
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Calculate $(g \bigcirc f)(-3)$ for $f(x) = 3x + 1$ and $g(x) = x^2$.
Calculate $(g \bigcirc f)(-3)$ for $f(x) = 3x + 1$ and $g(x) = x^2$.
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$(g \bigcirc f)(-3) = 64$. $f(-3) = -8$, then $g(-8) = 64$.
$(g \bigcirc f)(-3) = 64$. $f(-3) = -8$, then $g(-8) = 64$.
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If $f(x) = 4x$ and $g(x) = x - 2$, find $(f \bigcirc g)(x)$.
If $f(x) = 4x$ and $g(x) = x - 2$, find $(f \bigcirc g)(x)$.
Tap to reveal answer
$(f \bigcirc g)(x) = 4(x - 2)$. Apply $f$ to $g(x) = x - 2$: $4(x - 2)$.
$(f \bigcirc g)(x) = 4(x - 2)$. Apply $f$ to $g(x) = x - 2$: $4(x - 2)$.
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Find $(f \bigcirc g)(x)$ if $f(x) = x^2 + 1$ and $g(x) = 3x$.
Find $(f \bigcirc g)(x)$ if $f(x) = x^2 + 1$ and $g(x) = 3x$.
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$(f \bigcirc g)(x) = 9x^2 + 1$. Substitute $g(x) = 3x$ into $f$: $(3x)^2 + 1$.
$(f \bigcirc g)(x) = 9x^2 + 1$. Substitute $g(x) = 3x$ into $f$: $(3x)^2 + 1$.
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If $f(x) = x - 3$ and $g(x) = x^2$, find $(f \bigcirc g)(x)$.
If $f(x) = x - 3$ and $g(x) = x^2$, find $(f \bigcirc g)(x)$.
Tap to reveal answer
$(f \bigcirc g)(x) = x^2 - 3$. Apply $f$ to $g(x) = x^2$: $x^2 - 3$.
$(f \bigcirc g)(x) = x^2 - 3$. Apply $f$ to $g(x) = x^2$: $x^2 - 3$.
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Evaluate $(g \bigcirc f)(2)$ if $f(x) = x + 1$ and $g(x) = 2x$.
Evaluate $(g \bigcirc f)(2)$ if $f(x) = x + 1$ and $g(x) = 2x$.
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$(g \bigcirc f)(2) = 6$. $f(2) = 3$, then $g(3) = 6$.
$(g \bigcirc f)(2) = 6$. $f(2) = 3$, then $g(3) = 6$.
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Determine $(f \bigcirc g)(x)$ for $f(x) = x^2$ and $g(x) = x + 1$.
Determine $(f \bigcirc g)(x)$ for $f(x) = x^2$ and $g(x) = x + 1$.
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$(f \bigcirc g)(x) = (x + 1)^2$. Apply $f$ to $g(x) = x + 1$: $(x + 1)^2$.
$(f \bigcirc g)(x) = (x + 1)^2$. Apply $f$ to $g(x) = x + 1$: $(x + 1)^2$.
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What is $(f \bigcirc g)(0)$ for $f(x) = 2x + 3$ and $g(x) = x^2$?
What is $(f \bigcirc g)(0)$ for $f(x) = 2x + 3$ and $g(x) = x^2$?
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$(f \bigcirc g)(0) = 3$. $g(0) = 0$, then $f(0) = 3$.
$(f \bigcirc g)(0) = 3$. $g(0) = 0$, then $f(0) = 3$.
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If $f(x) = x^2$ and $g(x) = x + 2$, find $(g \bigcirc f)(x)$.
If $f(x) = x^2$ and $g(x) = x + 2$, find $(g \bigcirc f)(x)$.
Tap to reveal answer
$(g \bigcirc f)(x) = x^2 + 2$. Apply $g$ to $f(x) = x^2$: $x^2 + 2$.
$(g \bigcirc f)(x) = x^2 + 2$. Apply $g$ to $f(x) = x^2$: $x^2 + 2$.
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Find $(f \bigcirc g)(-1)$ if $f(x) = x^2$ and $g(x) = x + 5$.
Find $(f \bigcirc g)(-1)$ if $f(x) = x^2$ and $g(x) = x + 5$.
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$(f \bigcirc g)(-1) = 16$. $g(-1) = 4$, then $f(4) = 16$.
$(f \bigcirc g)(-1) = 16$. $g(-1) = 4$, then $f(4) = 16$.
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If $f(x) = 5x$ and $g(x) = x - 3$, find $(f \bigcirc g)(x)$.
If $f(x) = 5x$ and $g(x) = x - 3$, find $(f \bigcirc g)(x)$.
Tap to reveal answer
$(f \bigcirc g)(x) = 5(x - 3)$. Substitute $g(x) = x - 3$ into $f(x) = 5x$.
$(f \bigcirc g)(x) = 5(x - 3)$. Substitute $g(x) = x - 3$ into $f(x) = 5x$.
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Calculate $(g \bigcirc f)(3)$ for $f(x) = 2x$ and $g(x) = x^2$.
Calculate $(g \bigcirc f)(3)$ for $f(x) = 2x$ and $g(x) = x^2$.
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$(g \bigcirc f)(3) = 36$. $f(3) = 6$, then $g(6) = 36$.
$(g \bigcirc f)(3) = 36$. $f(3) = 6$, then $g(6) = 36$.
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If $f(x) = x^2$ and $g(x) = x + 1$, what is $(f \bigcirc g)(1)$?
If $f(x) = x^2$ and $g(x) = x + 1$, what is $(f \bigcirc g)(1)$?
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$(f \bigcirc g)(1) = 4$. $g(1) = 2$, then $f(2) = 4$.
$(f \bigcirc g)(1) = 4$. $g(1) = 2$, then $f(2) = 4$.
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What is $(f \bigcirc g)(x)$ for $f(x) = \frac{1}{2}x$ and $g(x) = x + 4$?
What is $(f \bigcirc g)(x)$ for $f(x) = \frac{1}{2}x$ and $g(x) = x + 4$?
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$(f \bigcirc g)(x) = \frac{1}{2}(x + 4)$. Apply $f$ to $g(x) = x + 4$.
$(f \bigcirc g)(x) = \frac{1}{2}(x + 4)$. Apply $f$ to $g(x) = x + 4$.
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What is the definition of the composition of functions?
What is the definition of the composition of functions?
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It is applying one function to the results of another: $(f \bigcirc g)(x) = f(g(x))$. First apply g, then apply f to that result.
It is applying one function to the results of another: $(f \bigcirc g)(x) = f(g(x))$. First apply g, then apply f to that result.
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State the notation for composing functions $f(x)$ and $g(x)$.
State the notation for composing functions $f(x)$ and $g(x)$.
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$(f \bigcirc g)(x) = f(g(x))$. Standard notation for function composition.
$(f \bigcirc g)(x) = f(g(x))$. Standard notation for function composition.
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