Objective 2 - Composition

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Evaluate the composition of two functions.

Link to section in online textbook.

First, watch this video to learn how use a new operation on functions: Composition.

For the two functions below, evaluate $(f\circ g)(\sage {valueQ1})$ and $(g \circ f)(\sage {valueQ1})$

$f(x) = \sage {functionQ1f}$ $g(x) = \sage {functionQ1g}$

$(f\circ g)(\sage {valueQ1}) = \answer [tolerance=0.05]{\sage {evaluateFGQ1}}$

$(g \circ f)(\sage {valueQ1}) = \answer [tolerance=0.05]{\sage {evaluateGFQ1}}$

Remember, the order is important! This joins subtraction and division where the order matters.

For the two functions below, evaluate $(f\circ g)(\sage {valueQ2})$ and $(g \circ f)(\sage {valueQ2})$

$f(x) = \sage {functionQ2f}$ $g(x) = \sage {functionQ2g}$

$(f\circ g)(\sage {valueQ2}) = \answer [tolerance=0.05]{\sage {evaluateFGQ2}}$

$(g \circ f)(\sage {valueQ2}) = \answer [tolerance=0.05]{\sage {evaluateGFQ2}}$

Great! Beyond our first question, we needed to be careful that we could plug in our values, as $g(x)$ has a restricted domain.

For the two functions below, evaluate $(f\circ g)(\sage {valueQ3})$ and $(g \circ f)(\sage {valueQ3})$

$f(x) = \sage {functionQ3f}$ $g(x) = \sage {functionQ3g}$

$(f\circ g)(\sage {valueQ3}) = \answer [tolerance=0.05]{\sage {evaluateFGQ3}}$

$(g \circ f)(\sage {valueQ3}) = \answer [tolerance=0.05]{\sage {evaluateGFQ3}}$

Great! Beyond our first question, we needed to be careful that we could plug in our values, as $g(x)$ has a restricted domain.

For the two functions below, evaluate $(f\circ g)(\sage {valueQ4})$ and $(g \circ f)(\sage {valueQ4})$

$f(x) = \sage {functionQ4f}$ $g(x) = \sage {functionQ4g}$

$(f\circ g)(\sage {valueQ4}) = \answer [tolerance=0.05]{\sage {evaluateFGQ4}}$

$(g \circ f)(\sage {valueQ4}) = \answer [tolerance=0.05]{\sage {evaluateGFQ4}}$

One of the biggest takeaways from this objective is noticing that $(f \circ g)(x) \neq (g \circ f)(x)$ in most cases.

For which question was $(f \circ g)(x) = (g \circ f)(x)$ ? $\answer {4}$

It is just asking for the number of the question. The answer is either “1”, “2”, “3”, or “4”.

Great job! But since it wasn’t always the case, it should make you wonder: when is it that $(f \circ g)(x) = (g \circ f)(x)$ ? The next two objectives will answer this question.