For
𝑓
and
𝑔
real valued functions defined for all real numbers,
(
𝑔
∘
𝑓
)
(
𝑥
)
=
𝑔
(
𝑓
(
𝑥
)
)
or all
𝑥
. Which of the following is (are) true?

I. If
𝑓
is increasing and
𝑔
is increasing, then
𝑔
∘
𝑓
is increasing.

II. If
𝑓
is increasing and
𝑔
is decreasing, then
𝑔
∘
𝑓
is decreasing.

III. If
𝑓
is decreasing and
𝑔
is increasing, then
𝑔
∘
𝑓
is decreasing.

Question

For 
𝑓
 and 
𝑔
 real valued functions defined for all real numbers, 
(
𝑔
∘
𝑓
)
(
𝑥
)
=
𝑔
(
𝑓
(
𝑥
)
)
 or all 
𝑥
.  Which of the following is (are) true?

I.  If 
𝑓
 is increasing and 
𝑔
 is increasing, then 
𝑔
∘
𝑓
 is increasing.

II.  If 
𝑓
 is increasing and 
𝑔
 is decreasing, then 
𝑔
∘
𝑓
 is decreasing.

III.  If 
𝑓
 is decreasing and 
𝑔
 is increasing, then 
𝑔
∘
𝑓
 is decreasing.

BlackTom AI · Accepted Answer

Let’s restate the problem to ensure we’re evaluating the right relationships: we have real-valued functions f and g defined for all real numbers, and we’re looking at the monotonicity of the composition g∘f. The options to evaluate are: - I. If f is increasing and g is increasing, then g∘f is increasing. - II. If f is increasing and g is decreasing, then g∘f is decreasing. - III. If f is decreasing and g is increasing, then g∘f is decreasing. Now, examine each scenario step by step. Option I: - Suppose x1 < x2. Since f is increasing, we have f(x1) < f(x2). - Since g is increasing, applying g to those values preserves order: g(f(x1)) < g(f(x2)). - Therefore, (g∘f)(x1) < (g∘f)(x2), which means g∘f is increasing. This option is consistent with the basic rule: the composition of two increasing functions is increasing. Option II: - Let x1 < x2. With f increasing, f(x1) < f(x2). - Now g is decreasing, so larger inputs yield smaller outputs: g(f(x1)) > g(f(x2)) because f(x1) < f(x2) and g reverses order. - Hence (g∘f)(x1) > (g∘f)(x2), showing that g∘f is decreasing. This aligns with the general principle that composing an increasing function with a decreasing function (on the inside) can flip the monotonicity. Option III: - Take x1 < x2. If f is decreasing, then f(x1) > f(x2). - Since g is increasing, larger inputs to g yield larger outputs: g(f(x1)) > g(f(x2)) because f(x1) > f(x2). - Consequently, (g∘f)(x1) > (g∘f)(x2), which means g∘f is decreasing in this case as well. Across these three scenarios, each analysis shows g∘f behaves as described: I yields increasing, II yields decreasing, and III yields decreasing. In other words, all three statements align with the standard monotonicity rules for function composition under the given conditions. A common pitfall to watch for is assuming that the inner or outer monotonicity alone dictates the result without accounting for the direction of the inequality preserved or flipped by each function involved. To summarize the intuitive takeaway: when the inner function f preserves or reverses order, and the outer function g preserves or reverses order correspondingly, the resulting order of g∘f follows the combined effect of those two monotonicities, producing increasing behavior when both are increasing, and decreasing behavior when the outer function reverses the order relative to the inner one, as illustrated in options II and III. No hidden exceptions arise from the domain here since both functions are defined for all real numbers, so the standard composition rules apply throughout.

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