show that the composition of two increasing functions is increasing
(fog)(x) = f(g(x))
f(x+h) > f(x) if h>0
g(x+h) > g(x) if h>0, so g(x+h) = g(x) + k for some value of k
consider f(g(x+h)). g(x+h) > g(x), so
f(g(x+h)) > f(g(x))
so, increasing
Oh, you want me to prove that the composition of two increasing functions is increasing? Well, get ready for some mathematical comedy!
Imagine two increasing functions, f(x) and g(x), walking into a bar. They decide to go for a drink together. As they sit down and order their drinks, f(x) thinks to itself, "I'm going to stay classy and always increase my value as x increases." And g(x) replies, "Hey, that sounds like a great idea. I'll do the same!"
Then, the bartender brings them their drinks and says, "Here you go, gentlemen. Enjoy!" The drinks are served in this special bar, where the glasses are labeled with the x-values. So, the glasses are lined up on the bar counter in increasing order.
Now, when f(x) takes a sip, it looks at its glass and says, "Hmm, my value is increasing, so let me increase it even more." Then, it takes another sip, and it does the same thing again. Each time, it keeps increasing its value.
Meanwhile, g(x) does the same thing with its own glass. It takes a sip, looks at its glass, and says, "My value is increasing too, so I'm going to increase it even further!" It takes another sip and increases its value.
But here's the funny part: when f(g(x)) takes a sip from g(x)'s glass, it says, "Hold on a second. I'm the composition of two increasing functions. That means I have to increase my value as well!"
So, f(g(x)) takes a sip, looks at the glass labeled with g(x)'s value, and says, "I'm going to increase it even more!" It takes another sip and increases its value too. And guess what? It keeps doing this every time it takes a sip from g(x)'s glass.
Therefore, the composition of f(x) and g(x), which is f(g(x)), is also an increasing function. Just like f(x) and g(x), it keeps increasing its value as x increases.
And that, my friend, is why the composition of two increasing functions is increasing. Cheers to math and comedy!
To show that the composition of two increasing functions is increasing, we need to assume that we have two increasing functions, let's call them f and g, and we want to prove that their composition f(g(x)) is also an increasing function.
Let's begin the proof:
1. Assume we have two increasing functions f(x) and g(x) defined on some interval I.
2. Since f(x) is increasing, for any two values a and b in I such that a < b, we have f(a) < f(b).
3. Similarly, since g(x) is increasing, for any two values c and d in I such that c < d, we have g(c) < g(d).
4. Now, let's consider the composition of f(g(x)), and let's take two values x1 and x2 in I such that x1 < x2.
5. We want to prove that f(g(x1)) < f(g(x2)).
To prove this, we will use the property of increasing functions:
6. Since f(x) is increasing, and for any x1 and x2 such that x1 < x2, we have f(g(x1)) < f(g(x2)), which holds because g(x1) < g(x2) based on the definition of an increasing function.
7. Therefore, we have shown that the composition of two increasing functions, f(g(x)), is also increasing.
By following these steps, we have successfully shown that the composition of two increasing functions is increasing.
To show that the composition of two increasing functions is increasing, we need to prove that if we have two increasing functions f and g, then the composition function f(g(x)) is also increasing.
To do this, we can use the definition of an increasing function. A function f is increasing if for any two values a and b, where a < b, we have f(a) < f(b).
Now, let's consider the composition function f(g(x)). Let's take two values, c and d, where c < d. We want to show that f(g(c)) < f(g(d)).
Since g is an increasing function, we know that g(c) < g(d) because c < d. Now, since f is an increasing function, we also know that f(g(c)) < f(g(d)), because g(c) < g(d).
Therefore, by using the definitions of increasing functions, we have shown that if f and g are increasing functions, then the composition f(g(x)) is also an increasing function.