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Formula for composite function ff of gg of xx.

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

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Formula for composite function ff of gg of xx.

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

What is the identity function?

f(x)=xf(x) = x

Formula for vertical translation up by kk units.

f(x)+kf(x) + k

Formula for vertical translation down by kk units.

f(x)kf(x) - k

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

f(kx)f(kx)

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

f(kx)f(kx)

Explain the order of operations in f(g(x))f(g(x)).

Evaluate g(x)g(x) first, then use the result as the input for f(x)f(x). Work from the inside out.

Why is the order important in composite functions?

Composition is generally not commutative; f(g(x))f(g(x)) is usually not equal to g(f(x))g(f(x)).

What happens when you compose a function with the identity function?

The result is the original function; f(x)=xf(x) = x implies g(f(x))=f(g(x))=g(x)g(f(x)) = f(g(x)) = g(x).

How do you find f(g(x))f(g(x)) analytically?

Substitute the entire function g(x)g(x) for every instance of xx in f(x)f(x).

How do composite functions relate to transformations?

They can represent transformations such as vertical translations (f(x)+kf(x) + k) and horizontal dilations (f(kx)f(kx)).

Explain how to use graphs to evaluate composite functions.

Find the output of g(x)g(x) from its graph, then use that output as the input for f(x)f(x) on its graph.

How to find f(g(x))f(g(x)) given f(x)=x+1f(x) = x + 1 and g(x)=x2g(x) = x^2?

Replace xx in f(x)f(x) with g(x)g(x): f(g(x))=(x2)+1=x2+1f(g(x)) = (x^2) + 1 = x^2 + 1.

How to evaluate f(g(2))f(g(2)) given f(x)=2x1f(x) = 2x - 1 and g(x)=x2+3g(x) = x^2 + 3?

First, find g(2)=22+3=7g(2) = 2^2 + 3 = 7. Then, find f(7)=2(7)1=13f(7) = 2(7) - 1 = 13.

How to decompose h(x)=(x+2)2h(x) = (x + 2)^2 into two functions, f(x)f(x) and g(x)g(x)?

Let g(x)=x+2g(x) = x + 2 and f(x)=x2f(x) = x^2. Then f(g(x))=(x+2)2=h(x)f(g(x)) = (x + 2)^2 = h(x).

How to find xx such that f(g(x))=5f(g(x)) = 5, given f(x)=x+1f(x) = x + 1 and g(x)=2xg(x) = 2x?

First find f(g(x))=2x+1f(g(x)) = 2x + 1. Then solve 2x + 1 = 5, which gives x=2x = 2.

How to find f(g(x))f(g(x)) if f(x)=x2f(x) = x^2 and g(x)=xg(x) = \sqrt{x}?

Substitute g(x)g(x) into f(x)f(x): f(g(x))=(x)2=xf(g(x)) = (\sqrt{x})^2 = x, for x0x \geq 0.

How to determine the domain of f(g(x))f(g(x))?

Find the domain of g(x)g(x) and ensure that the range of g(x)g(x) is within the domain of f(x)f(x).