given a general function f(x) and absolute value function a(x) = |x|, answer the given questions.
a. when is the graph of f(x) = a(f(x))? explain using complete sentences.
b. how is the graph of a(f(x)) related to the graph of f(x)? explain using complete sentences.
c. when is the graph of f(x) = f(a(x))? explain using complete sentences.
d. how is the graph of f(a(x)) related to the graph of f(x)? explain using complete sentences.
*bonus* given the graph below of f(x), graph y = |f(x)| and y = f(|x|) using different colors. describe any similarities and differences of the graphs.

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### Part a # Brief Explanations: To determine when $ f(x)=a(f(x)) $ (where $ a(x) = |x| $), we substitute $ a(f(x)) $ with $ |f(x)| $. So we need $ f(x)=|f(x)| $. By the definition of absolute value, $ |y| = y $ if and only if $ y\geq0 $. So this equation holds when $ f(x)\geq0 $ for all $ x $ in the domain of $ f $. In other words, the graph of $ f(x) $ lies on or above the $ x $-axis (has non - negative $ y $-values everywhere). # Answer: The graph of $ f(x)=a(f(x)) $ when $ f(x)\geq0 $ for all $ x $ in the domain of $ f(x) $ (i.e., the graph of $ f(x) $ is non - negative everywhere, lying on or above the $ x $-axis). ### Part b # Brief Explanations: We know that $ a(f(x))=|f(x)| $. For any point $ (x,y) $ on the graph of $ f(x) $, the corresponding point on the graph of $ |f(x)| $ is $ (x,|y|) $. If $ y\geq0 $ (the part of $ f(x) $ above or on the $ x $-axis), then $ |y| = y $, so this part of the graph remains unchanged. If $ y < 0 $ (the part of $ f(x) $ below the $ x $-axis), then $ |y|=-y $, which means we reflect the part of the graph of $ f(x) $ that is below the $ x $-axis over the $ x $-axis to get the graph of $ |f(x)| $. # Answer: The graph of $ a(f(x)) = |f(x)| $ is related to the graph of $ f(x) $ as follows: The portion of the graph of $ f(x) $ that lies on or above the $ x $-axis (where $ f(x)\geq0 $) remains the same. The portion of the graph of $ f(x) $ that lies below the $ x $-axis (where $ f(x)<0 $) is reflected over the $ x $-axis to form the graph of $ |f(x)| $. ### Part c # Brief Explanations: We want to find when $ f(x)=f(a(x)) $ with $ a(x)=|x| $, so we need $ f(x)=f(|x|) $ for all $ x $ in the domain. This means that the function $ f $ has the property that $ f(x)=f(-x) $ for all $ x $ in the domain. A function that satisfies $ f(x)=f(-x) $ for all $ x $ in its domain is called an even function. Geometrically, even functions are symmetric about the $ y $-axis. So the graph of $ f(x) $ is symmetric about the $ y $-axis (i.e., $ f $ is an even function). # Answer: The graph of $ f(x)=f(a(x)) $ when $ f(x) $ is an even function, that is, when $ f(x)=f(-x) $ for all $ x $ in the domain of $ f(x) $ (the graph of $ f(x) $ is symmetric about the $ y $-axis). ### Part d # Brief Explanations: We know that $ f(a(x))=f(|x|) $. Let's consider the domain of $ x $. For $ x\geq0 $, $ |x| = x $, so $ f(|x|)=f(x) $. This means that the part of the graph of $ f(|x|) $ for $ x\geq0 $ is the same as the part of the graph of $ f(x) $ for $ x\geq0 $. For $ x < 0 $, $ |x|=-x $, so $ f(|x|)=f(-x) $. Since $ x < 0 $, let $ t=-x>0 $, then $ f(-x)=f(t) $, which is the same as the value of $ f $ at $ t > 0 $. Geometrically, this means that we take the part of the graph of $ f(x) $ for $ x\geq0 $ and reflect it over the $ y $-axis to get the part of the graph of $ f(|x|) $ for $ x < 0 $. In other words, the graph of $ f(|x|) $ is symmetric about the $ y $-axis, and it is formed by taking the right - hand side (for $ x\geq0 $) of the graph of $ f(x) $ and reflecting it over the $ y $-axis to get the left - hand side (for $ x < 0 $). # Answer: The graph of $ f(a(x))=f(|x|) $ is related to the graph of $ f(x) $ as follows: The portion of the graph of $ f(|x|) $ for $ x\geq0 $ is identical to the portion of the graph of $ f(x) $ for $ x\geq0 $. The portion of the graph of $ f(|x|) $ for $ x < 0 $ is the reflection of the portion of the graph of $ f(x) $ for $ x>0 $ over the $ y $-axis. So the graph of $ f(|x|) $ is symmetric about the $ y $-axis, and it is constructed by using the right - half ( $ x\geq0 $) of $ f(x) $ and reflecting it to the left - half ( $ x < 0 $). ### Bonus #### Graphing $ y = |f(x)| $ - **For the given graph of $ f(x) $**: - Identify the parts of the graph of $ f(x) $ above and below the $ x $-axis. The parts above or on the $ x $-axis (where $ f(x)\geq0 $) stay the same. The parts below the $ x $-axis (where $ f(x)<0 $) are reflected over the $ x $-axis. For example, if we have a point $ (x,y) $ on $ f(x) $ with $ y < 0 $, the corresponding point on $ |f(x)| $ is $ (x, - y) $. - **Graphing $ y=f(|x|) $**: - For $ x\geq0 $, $ |x|=x $, so the graph of $ f(|x|) $ for $ x\geq0 $ is the same as the graph of $ f(x) $ for $ x\geq0 $. For $ x < 0 $, $ |x|=-x $, so we take the graph of $ f(x) $ for $ x > 0 $ and reflect it over the $ y $-axis to get the graph of $ f(|x|) $ for $ x < 0 $. #### Similarities and Differences - **Similarities**: - Both $ y = |f(x)| $ and $ y = f(|x|) $ are symmetric with respect to the $ y $-axis? No, $ y = |f(x)| $ is symmetric about the $ x $-axis? No, $ y = |f(x)| $ is symmetric about the $ x $-axis in the sense that if $ (x,y) $ is on $ |f(x)| $, then $ (x, - y) $ is not necessarily, but the reflection over $ x $-axis is how it's formed from $ f(x) $. Wait, actually, $ y = f(|x|) $ is always symmetric about the $ y $-axis because $ f(|-x|)=f(|x|) $. $ y = |f(x)| $ is symmetric about the $ x $-axis? No, $ |f(-x)|=|f(x)| $, so $ y = |f(x)| $ is also symmetric about the $ y $-axis. So both graphs are symmetric about the $ y $-axis. - **Differences**: - The transformation for $ y = |f(x)| $ affects the $ y $-values (reflection over $ x $-axis for negative $ y $-values of $ f(x) $), while the transformation for $ y = f(|x|) $ affects the $ x $-values (reflection over $ y $-axis for negative $ x $-values). For example, if $ f(x) $ has a part below the $ x $-axis, $ |f(x)| $ will reflect that part over $ x $-axis, but $ f(|x|) $ will not change that part (it will just mirror the right - hand side to the left - hand side). If $ f(x) $ is not even and has a non - symmetric part for $ x>0 $ and $ x < 0 $, $ |f(x)| $ will change the $ y $-values of the $ x < 0 $ part (if $ f(x) $ is negative there) while $ f(|x|) $ will mirror the $ x>0 $ part to $ x < 0 $ without changing the $ y $-values (just the $ x $-input to $ f $). If we take the given graph of $ f(x) $: - For $ y = |f(x)| $: - The parts of $ f(x) $ above the $ x $-axis (e.g., between $ x=-4 $ and $ x = - 2 $, and between $ x = 0 $ and $ x = 2 $, and $ x = 4 $) remain the same. The parts of $ f(x) $ below the $ x $-axis (e.g., between $ x=-2 $ and $ x = 0 $, and between $ x = 2 $ and $ x = 4 $) are reflected over the $ x $-axis. - For $ y = f(|x|) $: - The part of $ f(x) $ for $ x\geq0 $ (from $ x = 0 $ to $ x = 4 $) is taken, and then reflected over the $ y $-axis to get the part for $ x < 0 $. So the left - hand side ( $ x < 0 $) of $ f(|x|) $ is the mirror image of the right - hand side ( $ x>0 $) of $ f(x) $ over the $ y $-axis. The similarity is that both graphs are symmetric about the $ y $-axis. The difference is that $ y = |f(x)| $ modifies the $ y $-values (reflecting below - axis parts over $ x $-axis) while $ y = f(|x|) $ modifies the $ x $-values (reflecting right - hand side over $ y $-axis to get left - hand side) without changing the $ y $-values of the corresponding $ f $-outputs (except as a result of the input change). # Answer: ### Graphing: - **For $ y = |f(x)| $**: Keep the parts of $ f(x) $ above/below the $ x $-axis as is/reflect over $ x $-axis respectively. - **For $ y = f(|x|) $**: Keep the $ x\geq0 $ part of $ f(x) $, reflect $ x > 0 $ part over $ y $-axis for $ x < 0 $. ### Similarities and Differences: - **Similarities**: Both graphs are symmetric about the $ y $-axis. - **Differences**: $ y = |f(x)| $ reflects below - axis parts of $ f(x) $ over $ x $-axis (modifies $ y $-values for $ x $ where $ f(x)<0 $); $ y = f(|x|) $ reflects $ x>0 $ part of $ f(x) $ over $ y $-axis to $ x < 0 $ (modifies $ x $-input to $ f $, not $ y $-values directly).

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Part a

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Part b