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  1. AP Pre Calculus
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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.

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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).

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.

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).