# Let #f(x)=8x # and #g(x)=x/8#, how do you find each of the compositions and domain and range?

To find the composition of (f \circ g) (denoted as (f(g(x)))), substitute the expression for (g(x)) into (f(x)). Similarly, to find the composition of (g \circ f) (denoted as (g(f(x)))), substitute the expression for (f(x)) into (g(x)).

(f(g(x)) = f\left(\frac{x}{8}\right) = 8\left(\frac{x}{8}\right) = x)

(g(f(x)) = g(8x) = \frac{8x}{8} = x)

For the domain and range:

For (f(x) = 8x), the domain is all real numbers ((\mathbb{R})) because there are no restrictions on the values (x) can take. The range is also all real numbers ((\mathbb{R})) because for any input (x), (f(x)) produces an output that spans the entire real number line.

For (g(x) = \frac{x}{8}), the domain is again all real numbers ((\mathbb{R})), and the range is also all real numbers ((\mathbb{R})) because dividing any real number by 8 still results in a real number.

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Both functions and their compositions have Domains and Ranges of

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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