(b) if f(x) and g(x) are inverses, then f(g(x)) will equal what? (select all correct answer choices. there are more than one.) a f(g(x)) = -∞ b f(g(x)) = ∞ c f(g(x)) = x d f(g(x)) = y e f(g(x)) = g(f(x))

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# Brief Explanations:
To determine the correct answers when $ f(x) $ and $ g(x) $ are inverses, we use the definition of inverse functions. The composition of a function and its inverse, $ f(g(x)) $ and $ g(f(x)) $, should return the original input. Also, if $ y = f(x) $, then $ x = g(y) $, so $ f(g(x)) = x $ (since $ g(x) $ undoes what $ f $ does, and vice versa). Additionally, $ f(g(x)) = g(f(x)) $ because both compositions of inverse functions are equal to the identity function (which maps $ x $ to $ x $, and in some contexts, if we consider $ y $ as the output, but more accurately, the key property is $ f(g(x)) = x $ and $ g(f(x)) = x $, and also $ f(g(x)) = g(f(x)) $). Let's analyze each option:

- Option A: $ f(g(x)) = -\infty $ is incorrect because inverse functions' composition should return the input, not infinity.
- Option B: $ f(g(x)) = \infty $ is incorrect for the same reason as A.
- Option C: $ f(g(x)) = x $ is correct by the definition of inverse functions (the composition of a function and its inverse is the identity function, which maps $ x $ to $ x $).
- Option D: If we consider $ y = f(x) $, then $ x = g(y) $, so $ f(g(x)) = f(g(x)) $. If we let $ y = f(x) $, then $ g(y) = x $, so $ f(g(x)) = f(g(x)) $. Wait, actually, if $ y $ is the output of $ f(x) $, but more directly, if we think of $ f(g(x)) $, since $ g(x) $ is the inverse, $ f(g(x)) = x $, but sometimes $ y $ is used to represent the output, but in the context of inverse functions, the standard property is $ f(g(x)) = x $ and $ g(f(x)) = x $. However, if we consider that $ y = f(x) $ implies $ x = g(y) $, then $ f(g(x)) = x $, but if we substitute $ y $ as the variable, maybe in some contexts, but actually, the correct property is $ f(g(x)) = x $ and $ f(g(x)) = g(f(x)) $. Wait, let's re-express:

The definition of inverse functions: If $ f $ and $ g $ are inverses, then $ f(g(x)) = x $ for all $ x $ in the domain of $ g $, and $ g(f(x)) = x $ for all $ x $ in the domain of $ f $. Also, $ f(g(x)) = g(f(x)) $ because both equal $ x $.

Now, let's check Option D: $ f(g(x)) = y $. If $ y = f(x) $, then $ g(y) = x $, so $ f(g(x)) = f(g(x)) $. Wait, maybe this is a notation issue. If $ y $ is the output of $ f(x) $, then $ x = g(y) $, so $ f(g(x)) = f(g(x)) $. But actually, the standard correct properties are $ f(g(x)) = x $ and $ f(g(x)) = g(f(x)) $. Let's check Option E: $ f(g(x)) = g(f(x)) $ is correct because both $ f(g(x)) $ and $ g(f(x)) $ equal $ x $, so they are equal to each other.

Wait, maybe I made a mistake with Option D. Let's re-examine:

If $ f $ and $ g $ are inverses, then $ f(g(x)) = x $. If we let $ y = f(x) $, then $ x = g(y) $, so $ f(g(x)) = x $. But if we write $ f(g(x)) = y $, that would mean $ y = x $, which is consistent if $ y = x $, but that's a bit confusing. However, the key correct options are C and E, and maybe D? Wait, no, let's recall the exact definition.

The correct properties of inverse functions are:

1. $ f(g(x)) = x $ for all $ x $ in the domain of $ g $.
2. $ g(f(x)) = x $ for all $ x $ in the domain of $ f $.
3. $ f(g(x)) = g(f(x)) $ (since both equal $ x $).

Now, let's check each option again:

- Option C: $ f(g(x)) = x $ is correct (by definition).
- Option D: $ f(g(x)) = y $. If $ y $ is the output of $ f(x) $, then $ y = f(x) $, so $ g(y) = x $, so $ f(g(x)) = f(g(x)) $. But if we consider $ y $ as a variable, and if $ f(g(x)) = x $, and if $ y = x $, then $ f(g(x)) = y $ would be true (since $ x = y $ in that case). But this is a bit ambiguous. However, the standard correct options are C and E, and maybe D? Wait, no, let's think again.

Wait, the problem says "Select ALL correct answer choices". Let's re-express:

- Option C: $ f(g(x)) = x $ is correct (identity function property).
- Option E: $ f(g(x)) = g(f(x)) $ is correct because both $ f(g(x)) $ and $ g(f(x)) $ equal $ x $, so they are equal to each other.
- Option D: $ f(g(x)) = y $. If we consider $ y $ as the output of $ f(x) $, then $ y = f(x) $, so $ g(y) = x $, so $ f(g(x)) = f(g(x)) $. But if we take $ y = x $ (since $ f(g(x)) = x $), then $ f(g(x)) = y $ would be true (because $ x = y $). So maybe D is also correct? Wait, this is confusing. Let's check the definitions again.

The definition of inverse functions: If $ f $ and $ g $ are inverses, then $ f(g(x)) = x $ and $ g(f(x)) = x $ for all $ x $ in their respective domains. Also, $ f(g(x)) = g(f(x)) $ because both equal $ x $. Now, if we let $ y = f(x) $, then $ x = g(y) $, so $ f(g(x)) = f(g(x)) $. But if we substitute $ y = x $ (since $ f(g(x)) = x $), then $ f(g(x)) = y $ (because $ y = x $). So in that case, D is also correct? Wait, maybe the problem uses $ y $ to represent the output, so if $ f(g(x)) = x $, and if $ y = x $, then $ f(g(x)) = y $. So maybe C, D, and E are correct? Wait, no, let's check the options again:

- Option C: $ f(g(x)) = x $ – correct.
- Option D: $ f(g(x)) = y $ – if $ y = x $, then this is correct (since $ f(g(x)) = x = y $ if $ y = x $).
- Option E: $ f(g(x)) = g(f(x)) $ – correct, because both equal $ x $.

Wait, but the standard answer for inverse functions' composition is $ f(g(x)) = x $ and $ g(f(x)) = x $, and also $ f(g(x)) = g(f(x)) $. So let's confirm:

- Option C: Correct (by definition of inverse functions: $ f(g(x)) = x $).
- Option E: Correct (since $ f(g(x)) = x $ and $ g(f(x)) = x $, so $ f(g(x)) = g(f(x)) $).
- Option D: If we consider $ y $ as the output, and since $ f(g(x)) = x $, and if $ y $ is used to represent $ x $ (the input/output), then $ f(g(x)) = y $ (where $ y = x $) – this is a bit of a notation issue, but maybe it's considered correct. However, the most definite correct options are C and E, and possibly D. Wait, but let's check with an example. Let $ f(x) = 2x $, then $ g(x) = \frac{x}{2} $. Then $ f(g(x)) = f(\frac{x}{2}) = 2(\frac{x}{2}) = x $, and $ g(f(x)) = g(2x) = \frac{2x}{2} = x $. So $ f(g(x)) = x $, $ g(f(x)) = x $, so $ f(g(x)) = g(f(x)) $. Now, if we let $ y = f(x) = 2x $, then $ g(y) = \frac{y}{2} = x $, so $ f(g(x)) = x $, and if we take $ y = x $ (which is not the case here, because $ y = 2x $), then $ f(g(x)) = y $ would mean $ x = y $, which is only true if $ x = 2x $ (i.e., $ x = 0 $), which is not general. So maybe Option D is incorrect. Wait, maybe the problem has a typo or uses $ y $ to represent $ x $. Alternatively, maybe the intended correct options are C and E.

Wait, the problem says "Select ALL correct answer choices". Let's re-express the options:

- A: $ f(g(x)) = -\infty $ – incorrect.
- B: $ f(g(x)) = \infty $ – incorrect.
- C: $ f(g(x)) = x $ – correct (by definition of inverse functions: $ f(g(x)) = x $).
- D: $ f(g(x)) = y $ – if $ y $ is the output, but the output of $ f(g(x)) $ is $ x $, so if $ y = x $, then this is correct, but that's a bit of a stretch. However, in some contexts, $ y $ is used to represent the dependent variable, so if $ f(g(x)) = x $, and $ y = x $, then $ f(g(x)) = y $. But this is ambiguous.
- E: $ f(g(x)) = g(f(x)) $ – correct, because both $ f(g(x)) $ and $ g(f(x)) $ equal $ x $, so they are equal to each other.

Given that, the correct options are C and E, and possibly D. But let's check with the definition again. The key properties are:

1. $ f(g(x)) = x $ (identity function)
2. $ g(f(x)) = x $ (identity function)
3. $ f(g(x)) = g(f(x)) $ (since both equal $ x $)

So Option C is correct (matches property 1), Option E is correct (matches property 3), and Option D: If we consider $ y $ as the variable, and since $ f(g(x)) = x $, then $ f(g(x)) = y $ implies $ y = x $, which is a valid substitution (letting $ y = x $), so maybe D is also correct. But in standard math, the primary correct options for the composition of inverse functions are $ f(g(x)) = x $ and $ g(f(x)) = x $, and $ f(g(x)) = g(f(x)) $. So likely, the correct options are C, D, and E? Wait, no, in the example I gave, $ f(g(x)) = x $, and $ g(f(x)) = x $, so $ f(g(x)) = g(f(x)) $ (Option E is correct), $ f(g(x)) = x $ (Option C is correct), and if $ y = x $, then $ f(g(x)) = y $ (Option D is correct). So maybe all three C, D, E are correct. But let's check the options again:

- C: $ f(g(x)) = x $ – correct.
- D: $ f(g(x)) = y $ – if $ y = x $, then correct.
- E: $ f(g(x)) = g(f(x)) $ – correct, since both equal $ x $.

So the correct options are C, D, and E. Wait, but maybe the problem considers $ y $ as the output of $ f(x) $, so if $ f(x) = y $, then $ g(y) = x $, so $ f(g(x)) = f(g(x)) $. But in that case, $ f(g(x)) = x $, and if $ y = f(x) $, then $ x = g(y) $, so $ f(g(x)) = x = g(y) $, but that's not directly $ y $. Hmm, this is a bit confusing. But based on the definition, the definite correct options are C and E, and D might be correct depending on notation. However, in most standard problems, the correct options for the composition of inverse functions are $ f(g(x)) = x $ and $ f(g(x)) = g(f(x)) $, so C and E. Additionally, if $ y $ is used to represent $ x $ (the input/output), then D is also correct. But to be safe, the most solid correct options are C and E, and possibly D. But let's check the original problem again. The problem says "Select ALL correct answer choices. There are more than one." So let's confirm:

- Option C: $ f(g(x)) = x $ – correct (by definition of inverse functions: $ f \circ g = \text{id} $, where $ \text{id}(x) = x $).
- Option E: $ f(g(x)) = g(f(x)) $ – correct, because $ f \circ g = \text{id} $ and $ g \circ f = \text{id} $, so $ f \circ g = g \circ f $.
- Option D: $ f(g(x)) = y $ – if $ y $ is the variable representing the output, and since $ f(g(x)) = x $, then $ y = x $, so $ f(g(x)) = y $ is correct (substituting $ y = x $).

So the correct options are C, D, and E. Wait, but in the example I gave, $ f(g(x)) = x $, and if $ y = x $, then $ f(g(x)) = y $ is true. So yes, D is also correct. Therefore, the correct answer choices are C, D, and E.
# Answer:
C. $ f(g(x)) = x $
D. $ f(g(x)) = y $
E. $ f(g(x)) = g(f(x)) $

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