Problem 13 - Calculus - AP Edition

Question

Suppose that \(f\) and \(g\) are increasing functions. Determine which of the functions \(f(x)+g(x), f(x) g(x),\) and \(f(g(x))\) must also be increasing.

Step-by-Step Solution

Answer

\(f(x) + g(x)\) and \(f(g(x))\) are increasing. \(f(x)g(x)\) may not be.

Step 1: Understand the properties of increasing functions

A function \(h(x)\) is increasing if \(x_1 < x_2\) implies \(h(x_1) < h(x_2)\). This means that as \(x\) gets larger, the output of the function also gets larger.

Step 2: Analyze the function \(f(x) + g(x)\)

Since both \(f(x)\) and \(g(x)\) are increasing, for \(x_1 < x_2\), we have \(f(x_1) < f(x_2)\) and \(g(x_1) < g(x_2)\). Therefore, \(f(x_1) + g(x_1) < f(x_2) + g(x_2)\). Thus, \(f(x) + g(x)\) is also an increasing function.

Step 3: Analyze the function \(f(x)g(x)\)

To determine if \(f(x)g(x)\) is increasing, consider \(x_1 < x_2\). Since \(f(x_1) < f(x_2)\) and \(g(x_1) < g(x_2)\), both terms \(f(x_1)g(x_1)\) and \(f(x_2)g(x_2)\) aren't necessarily comparative directly unless both \(f(x)\) and \(g(x)\) are also positive. Without knowing if \(f(x)\) and \(g(x)\) are positive, we can't conclude that \(f(x)g(x)\) is increasing.

Step 4: Analyze the function \(f(g(x))\)

Start with \(x_1 < x_2\). Since \(g\) is increasing, \(g(x_1) < g(x_2)\). Also, since \(f\) is increasing, this means \(f(g(x_1)) < f(g(x_2))\). Therefore, \(f(g(x))\) is also an increasing function.

Key Concepts

Function Composition

Function composition is when you combine two functions in such a way that the output from one function becomes the input to another. Mathematically, this concept is represented as \(f(g(x))\). If \(f\) and \(g\) are both increasing functions, it generally means that an increase in the input \(x\) results in an increase in \(g(x)\), and subsequently, a further increase in \(f(g(x))\).

Let's break that down:

• When \(x_1 < x_2\), \(g(x_1) < g(x_2)\) because \(g\) is increasing.
• Subsequently, because \(f\) is also increasing, \(f(g(x_1)) < f(g(x_2))\).

Therefore, when both functions are increasing individually, their composition \(f(g(x))\) retains this property, making \(f(g(x))\) an increasing function. This means we can confidently say that the composite function is also increasing as \(x\) gets larger.

Function Addition

When we talk about function addition, we're referring to the process of adding two functions together. If \(f(x)\) and \(g(x)\) are functions, their sum is represented as \(f(x) + g(x)\). If both functions \(f\) and \(g\) are increasing, this property directly influences their sum.

Here's why:

• For \(x_1 < x_2\), \(f(x_1) < f(x_2)\) and \(g(x_1) < g(x_2)\).
• Therefore, when we add these inequalities, \(f(x_1) + g(x_1) < f(x_2) + g(x_2)\).

The result is an increasing function \(f(x) + g(x)\). This means that when you add two increasing functions together, the sum will also be increasing. Function addition honors the increasing nature when both original functions are increasing.

Function Multiplication

In function multiplication, we consider the product of two functions, represented by \(f(x)g(x)\). Whether their product is also increasing largely depends on the positivity of the functions involved.

If both \(f(x)\) and \(g(x)\) are always positive and increasing, then their product \(f(x)g(x)\) might also be increasing. Here's a detailed look:

• Given \(x_1 < x_2\), their increasing nature implies \(f(x_1) < f(x_2)\) and \(g(x_1) < g(x_2)\).
• If both functions are positive, then \(f(x_1)g(x_1) < f(x_2)g(x_2)\).

However, if \(f(x)\) and/or \(g(x)\) produce negative values, the product may not remain increasing due to the effect of negative multiplication. Without additional constraints guaranteeing positivity, we cannot conclusively say \(f(x)g(x)\) is increasing. Therefore, it's important to check the positivity condition before establishing that their multiplication is increasing.