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

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

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

$f(g(x))$ : Apply $g$ first, then $f$ . | $g(f(x))$ : Apply $f$ first, then $g$ . 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.

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

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

Why is the order important in composite functions?

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

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

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

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

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

How do composite functions relate to transformations?

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

Explain how to use graphs to evaluate composite functions.

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

Formula for composite function $f$ of $g$ of $x$ .

$f(g(x))$

What is the identity function?

$f(x) = x$

Formula for vertical translation up by $k$ units.

$f(x) + k$

Formula for vertical translation down by $k$ units.

$f(x) - k$

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

$f(kx)$

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

$f(kx)$

All Flashcards

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

$f(g(x))$ : Apply $g$ first, then $f$ . | $g(f(x))$ : Apply $f$ first, then $g$ . 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.

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

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

Why is the order important in composite functions?

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

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

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

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

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

How do composite functions relate to transformations?

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

Explain how to use graphs to evaluate composite functions.

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

Formula for composite function $f$ of $g$ of $x$ .

$f(g(x))$

What is the identity function?

$f(x) = x$

Formula for vertical translation up by $k$ units.

$f(x) + k$

Formula for vertical translation down by $k$ units.

$f(x) - k$

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

$f(kx)$

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

$f(kx)$