4 Operations on Functions
In the previous chapter Essentials of Functions, we explored the foundational concepts of functions—how they relate inputs to outputs, their domains and ranges, and the different types such as linear, quadratic, exponential, and logarithmic functions. Understanding these basic properties allows us to describe mathematical and real-world relationships systematically.
In this chapter, we extend that understanding by studying Operations on Functions, which involve combining two or more functions to create a new one. These operations allow us to model more complex relationships, solve optimization problems, and represent multi-step processes commonly found in engineering, business, and natural sciences. Essentially, operations on functions form the bridge between simple mathematical expressions and real-world systems modeling. Through addition, subtraction, multiplication, division, and composition of functions, we can represent how different processes interact with one another — for example, how production cost depends on both labor efficiency and material usage, or how temperature change affects reaction rates in thermodynamics.
This Figure 4.1 will cover:
- The definition of operations on functions
- The types of operations, including addition, subtraction, multiplication, division, and composition
- The purpose and importance of using these operations in modeling and analysis
- Applications in various fields such as engineering, mining, business, and metallurgy
By the end of this chapter, you will be able to combine functions effectively and interpret their interactions within real-world contexts.
4.1 Definition
Operations on functions involve combining two or more functions to create a new one, much like mixing ingredients to make a new recipe. Imagine you have two machines — one that squeezes oranges into juice and another that adds sugar and ice. When you connect them, you get a new machine that produces sweet orange juice. Likewise, in mathematics, when two or more functions are combined (through addition, subtraction, multiplication, division, or composition), they create a new function with new behavior and properties.
4.2 Types of Operations
Mathematical operations on functions allow us to combine or manipulate functions to form new ones. Functions can be combined or manipulated in several ways to form new functions. These operations are essential in modeling, analysis, and problem-solving in mathematics, science, engineering, and business.
4.2.1 Addition
Addition is a fundamental operation on functions, where the outputs of two functions are combined for the same input to produce a new function. Think of it like this: imagine two machines one makes orange juice \(f(x)\) and the other makes sugar syrup \(g(x)\). By combining them, you get a new machine producing sweet orange juice, which reflects the sum of their individual outputs.
Suppose we have two functions:
\[\begin{equation}\label{eq:add1} \begin{aligned} f(x) &= 2x + 3,\\ g(x) &= x^2. \end{aligned} \end{equation}\]
To find the sum of the two functions \(\eqref{eq:add1}\), we add their corresponding expressions as follows:
\[\begin{equation}\label{eq:add2} \begin{aligned} (f + g)(x) &= f(x) + g(x) \\ &= (2x + 3) + x^2 \\ &= x^2 + 2x + 3 \end{aligned} \end{equation}\]
Equation \(\eqref{eq:add2}\) shows that adding two functions produces a new function that combines the effects of both.
To gain a clearer understanding of the process of function addition, the following visualization illustrates how the functions \(f(x) = 2x + 3\) and \(g(x) = x^2\) interact. By plotting both functions along with their sum \((f + g)(x)\), we can observe how the resulting function combines the linear growth of \(f(x)\) with the quadratic behavior of \(g(x)\).
4.2.2 Subtraction
Subtraction is similar to addition but involves finding the difference between the outputs of two functions for the same input. Analogy: imagine the orange juice machine \(f(x)\) and the sugar syrup machine \(g(x)\). Subtracting them gives a machine producing juice with reduced sweetness, reflecting the difference of their contributions.
Suppose we have two functions:
\[\begin{equation}\label{eq:add3} \begin{aligned} f(x) = 5x,\\ g(x) = 2x. \end{aligned} \end{equation}\]
To find the difference of the two functions \(\eqref{eq:add3}\):, we subtract their corresponding expressions as follows:
\[\begin{equation}\label{eq:add4} \begin{aligned} (f - g)(x) &= f(x) - g(x) \\ &= 5x - 2x \\ &= 3x \end{aligned} \end{equation}\]
Equation \(\eqref{eq:add4}\) shows that the difference between two functions produces a new function that represents the subtraction of \(g(x)\) from \(f(x)\).
Before visualizing the subtraction of two functions, it is essential to understand that this operation involves determining the difference between their respective output values for each corresponding input. This process reveals how one function behaves relative to another, allowing us to analyze the resulting change or rate of difference between them.
4.2.3 Multiplication
Multiplication combines two functions by multiplying their outputs for the same input. Analogy: imagine the orange juice machine \(f(x)\) and a sugar syrup machine \(g(x)\). Multiplying them produces a machine that outputs the product of juice and syrup concentration, enhancing the effect multiplicatively.
Suppose we have two functions:
\[\begin{equation}\label{eq:add5} \begin{aligned} f(x) = x + 1,\\ g(x) = 2x. \end{aligned} \end{equation}\]
To find the product of the two functions \(\eqref{eq:add5}\), we multiply their corresponding expressions as follows:
\[\begin{equation}\label{eq:add6} \begin{aligned} (f \cdot g)(x) &= f(x) \cdot g(x) \\ &= (x + 1)(2x) \\ &= 2x^2 + 2x \end{aligned} \end{equation}\]
Equation \(\ref{eq:add6}\) shows that multiplying two functions combines their outputs multiplicatively, producing a new function that scales both effects together.
Before visualizing the multiplication of two functions, it is important to note that this operation involves combining the output values of both functions through pointwise multiplication.
The resulting product function illustrates how the interaction between the two functions amplifies or reduces their respective magnitudes across the domain.
4.2.4 Division
Division forms a new function by dividing the output of one function by another, provided the denominator is not zero. Analogy: imagine the orange juice machine \(f(x)\) and a syrup machine \(g(x)\). Dividing them produces a machine that controls sweetness ratio, balancing the two contributions.
Suppose we have two functions:
\[\begin{equation}\label{eq:add7} \begin{aligned} f(x) = x^2 + 4,\\ g(x) = 2x. \end{aligned} \end{equation}\]
To find the quotient of the two functions \(\ref{eq:add7}\), we divide their corresponding expressions as follows:
\[\begin{equation}\label{eq:add8} \begin{aligned} \left(\frac{f}{g}\right)(x) &= \frac{f(x)}{g(x)}, \quad g(x) \neq 0 \\[6pt] &= \frac{x^2 + 4}{2x} \\[6pt] &= \frac{x^2}{2x} + \frac{4}{2x} \\[6pt] &= \frac{x}{2} + \frac{2}{x}, \quad x \neq 0 \end{aligned} \end{equation}\]
Equation \(\ref{eq:add8}\) shows that dividing two functions produces a new function that represents the ratio of \(f(x)\) to \(g(x)\), provided \(g(x) \neq 0\).
Division of functions involves determining the ratio of two corresponding outputs, provided that the denominator function is non-zero. This operation produces a new function that illustrates how the numerator’s growth rate compares to that of the denominator across the domain.
4.2.5 Composition
Composition involves using the output of one function as the input of another. Analogy: imagine the orange juice machine \(g(x)\) feeds its juice into a blender \(f(x)\). The new machine produces a blended juice whose output depends on both machines in sequence.
Suppose we have two functions:
\[\begin{equation}\label{eq:add9} \begin{aligned} f(x) = x + 1,\\ g(x) = x^2. \end{aligned} \end{equation}\]
To find the composition of the two functions \(\ref{eq:add9}\), we substitute \(g(x)\) into \(f(x)\) as follows:
\[\begin{equation}\label{eq:add10} \begin{aligned} (f \circ g)(x) &= f(g(x)) \\ &= g(x) + 1 \\ &= x^2 + 1 \end{aligned} \end{equation}\]
Equation \(\ref{eq:add10}\) shows that the composition \((f \circ g)(x)\) produces a new function where the output of \(g(x)\) becomes the input of \(f(x)\).
Function composition involves applying one function to the result of another. In this case, the output of \(g(x)\) becomes the input to \(f(x)\), forming a new composite function \((f \circ g)(x) = f(g(x))\). This operation demonstrates how sequential transformations affect the shape of the resulting function.
4.3 Real-World Applications
4.3.1 A Car Engine
A car engine generates heat while operating. Without a cooling system, the engine temperature rises continuously, which can damage components. We want to model the actual engine temperature considering the effect of the cooling system. This is a real example of combining functions, where multiple factors are represented in a single equation.
Function Definitions
- Engine Temperature without Cooling
The engine generates heat over time: \(f(t) = 2t^2 + 30\)
- \(t\): time the engine runs (minutes)
- \(f(t)\): engine temperature (°C) without cooling
- Interpretation: temperature rises quadratically as heat increases faster over time.
- Cooling System Efficiency
The cooling system reduces heat, but efficiency decreases over time: \(g(t) = 0.5t + 1\)
- \(g(t)\): cooling factor (larger = more effective)
- As \(t\) increases, cooling efficiency becomes relatively weaker compared to engine heat.
Combining Functions
The actual engine temperature \(T(t)\) can be modeled as engine temperature divided by cooling efficiency:
\(T(t) = \frac{f(t)}{g(t)}\)
Substitute the functions:
\(T(t) = \frac{2t^2 + 30}{0.5t + 1}\)
Simplification
Step by step:
Factor out constants where possible: \(T(t) = \frac{2(t^2 + 15)}{0.5t + 1}\)
Multiply numerator and denominator by 2 to simplify: \(T(t) = \frac{4(t^2 + 15)}{t + 2}\)
Domain restriction: \(g(t) \neq 0 \Rightarrow t \neq -2\) (physically irrelevant since \(t \ge 0\)).
Interpretation
- At small \(t\), cooling is effective → engine temperature remains moderate.
- As \(t\) increases, the effect of cooling diminishes relative to engine heat → temperature rises faster.
- This model shows the interaction between two factors (engine heat and cooling efficiency) mathematically, not just from empirical observation.
Applications
- Cooling System Design: Estimate the cooling capacity needed to maintain safe temperatures.
- Engine Failure Prediction: Determine when engine temperature may become too high.
- Simulation & Optimization: Test various operation times, efficiency, or enhanced cooling without real-world experiments.
4.3.2 Energy and Metallurgy
In metallurgy, energy efficiency and chemical reactions are critical for designing furnaces and reactors. By combining functions, engineers can model heat generation, fuel consumption, and reaction progress, and optimize for efficiency or yield.
Combustion in a Blast Furnace
- Heat Generated by Fuel Combustion
The heat produced \(Q_f\) depends on the amount of fuel burned \(m_f\):
\(Q_f(m_f) = 30 m_f\) (MJ/kg of fuel)
- \(m_f\): mass of fuel burned (kg)
- \(Q_f\): total heat generated (MJ)
- Heat Required for Ore Reduction
The heat needed to reduce iron ore \(Q_r\) depends on the mass of ore \(m_o\):
\(Q_r(m_o) = 25 m_o\) (MJ/kg of ore)
- \(m_o\): mass of ore processed (kg)
- \(Q_r\): energy required for the reduction reaction (MJ)
Energy Balance and Efficiency
The process efficiency \(\eta\) can be modeled as the ratio of useful energy to energy supplied:
\(\eta(m_f, m_o) = \frac{Q_r(m_o)}{Q_f(m_f)} = \frac{25 m_o}{30 m_f}\)
- To achieve 100% energy efficiency, set \(\eta = 1\): \(25 m_o = 30 m_f \implies m_f = \frac{25}{30} m_o = 0.833 m_o\)
- This means for every 1 kg of ore, we need 0.833 kg of fuel for an ideal energy balance.
Interpretation
- \(\eta < 1\): fuel is overused → excess heat, higher cost.
- \(\eta > 1\): fuel is insufficient → reaction incomplete, lower yield.
- By combining \(Q_f\) and \(Q_r\) functions, we can optimize fuel usage and improve furnace efficiency.
Applications
- Metallurgy: Optimize fuel consumption for smelting or reduction reactions.
- Chemical Engineering: Balance energy inputs for industrial reactors.
- Sustainable Design: Reduce fuel costs and emissions by maximizing efficiency.
4.3.3 Others
Functions are widely used to model real-world systems. Often, we need to combine multiple functions to understand how different factors interact. The Table 4.1 provides practical examples for each type of function operation—addition, subtraction, multiplication, division, and composition—along with their interpretations.
Operation | Mathematical_Form | Real_World_Example | Description |
---|---|---|---|