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  1. AP Pre Calculus
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What is a composite function?

A function formed by applying one function to the results of another: f(g(x))f(g(x))f(g(x)).

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What is a composite function?

A function formed by applying one function to the results of another: f(g(x))f(g(x))f(g(x)).

What does f(g(x))f(g(x))f(g(x)) mean?

Apply ggg to xxx first, then apply fff to the result.

What is the identity function?

The function f(x)=xf(x) = xf(x)=x, which returns the input unchanged.

Define function decomposition.

Breaking down a complex function into simpler component functions.

What is vertical translation in terms of function composition?

Shifting the graph of a function up or down by adding a constant, represented by f(x)+kf(x) + kf(x)+k.

What is horizontal dilation in terms of function composition?

Stretching or shrinking the graph of a function horizontally by multiplying the input by a constant, represented by f(kx)f(kx)f(kx).

Formula for composite function fff of ggg of xxx.

f(g(x))f(g(x))f(g(x))

What is the identity function?

f(x)=xf(x) = xf(x)=x

Formula for vertical translation up by kkk units.

f(x)+kf(x) + kf(x)+k

Formula for vertical translation down by kkk units.

f(x)−kf(x) - kf(x)−k

Formula for horizontal dilation (stretch) by a factor of kkk where 0<k<10 < k < 10<k<1.

f(kx)f(kx)f(kx)

Formula for horizontal dilation (shrink) by a factor of kkk where k>1k > 1k>1.

f(kx)f(kx)f(kx)

What are the differences between f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))?

f(g(x))f(g(x))f(g(x)): Apply ggg first, then fff. | g(f(x))g(f(x))g(f(x)): Apply fff first, then ggg. The results are generally different.

Compare vertical translation and horizontal dilation.

Vertical Translation: Shifts the graph up or down. | Horizontal Dilation: Stretches or shrinks the graph horizontally.