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  1. AP Pre Calculus
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Explain the order of operations in f(g(x))f(g(x))f(g(x)).

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

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

Evaluate g(x)g(x)g(x) first, then use the result as the input for f(x)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))f(g(x)) is usually not equal to g(f(x))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) = xf(x)=x implies g(f(x))=f(g(x))=g(x)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))f(g(x)) analytically?

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

How do composite functions relate to transformations?

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

Explain how to use graphs to evaluate composite functions.

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

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).

How to find f(g(x))f(g(x))f(g(x)) given f(x)=x+1f(x) = x + 1f(x)=x+1 and g(x)=x2g(x) = x^2g(x)=x2?

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

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

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

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

Let g(x)=x+2g(x) = x + 2g(x)=x+2 and f(x)=x2f(x) = x^2f(x)=x2. Then f(g(x))=(x+2)2=h(x)f(g(x)) = (x + 2)^2 = h(x)f(g(x))=(x+2)2=h(x).

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

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

How to find f(g(x))f(g(x))f(g(x)) if f(x)=x2f(x) = x^2f(x)=x2 and g(x)=xg(x) = \sqrt{x}g(x)=x​?

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

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

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