Summary Tradisional | Exponential Function: Inputs and Outputs

Contextualization

Mathematical functions are vital instruments that assist us in modelling and understanding a wide range of phenomena happening in the world around us. Exponential functions, in particular, are commonly used to describe situations where something grows or shrinks at a rate that corresponds to its current value. Typical examples include population growth, the spread of diseases, radioactive decay, and even the calculation of compound interest in the financial sector.

For instance, in the context of social media, the rise in users on a platform like Instagram can be modelled by an exponential function. As more people sign up and invite others, the number of new users surges rapidly. Grasping these functions enables us to forecast trends and make informed decisions across various fields, from public health to economics.

To Remember!

Definition of Exponential Function

An exponential function is a type of mathematical function that takes the form f(x) = a * b^x, where 'a' is a non-zero coefficient, 'b' is the base (with b > 0 and b ≠ 1), and 'x' is the exponent. The base 'b' must be a positive constant different from 1 for the exponential function to display exponential growth or decay properties. The coefficient 'a' can affect the amplitude of the function but does not influence its growth or decay rate.

These functions are termed exponential because the exponent, 'x', varies while the base 'b' remains constant. This feature results in exponential growth or decay, which occurs much more swiftly than linear growth. For example, on a graph, an exponential function with b > 1 increases much faster than a linear function as x goes up.

The exponential function models phenomena where the rate of growth or decline is proportional to the current value. This can be seen in scenarios such as population growth, radioactive decay, and disease spread, where rapid changes happen because of the exponential nature of these situations.

• General form: f(x) = a * b^x

• Base 'b' is a positive constant different from 1

• Non-zero coefficient 'a' changes amplitude

Graph of Exponential Functions

The graph of an exponential function is distinctive, characterized by a curve that either grows or decays exponentially. When the base 'b' is greater than 1 (b > 1), the function increases rapidly as x gets larger. Conversely, when the base is between 0 and 1 (0 < b < 1), the function declines exponentially as x increases.

A key feature of the graph of exponential functions is that it never touches the x-axis. This indicates that the function never actually reaches zero but can get infinitely close to zero for negative values of x, given b > 1. For 0 < b < 1, the function approaches zero for positive values of x.

Examining the graph of exponential functions allows us to spot rapid changes in values that either grow or decline. These traits are vital for numerous practical applications, such as investigating population growth or the reduction of a radioactive substance over time.

• Exponential growth: b > 1

• Exponential decay: 0 < b < 1

• The function never touches the x-axis

Behavior of the Exponential Function

The behaviour of an exponential function fluctuates depending on the values of x. For positive values of x and base b > 1, the function expands quickly. This suggests that small increases in x result in significant increases in the output y. For negative values of x, the function tends to inch close to zero but never actually hits it.

When dealing with bases between 0 and 1 (0 < b < 1), the behaviour is the opposite, as the function quickly declines as x increases. This is useful for modelling scenarios that involve exponential decay, such as the decrease of a radioactive substance. For negative values of x, the function tends toward positive infinity, reflecting exponential growth backward in time.

Grasping this behaviour is crucial for correctly applying exponential functions in practical situations. By understanding how the function reacts to different values of x, one can make precise predictions and informed choices across various arenas.

• Rapid growth for b > 1 with positive x

• Rapid decay for 0 < b < 1 with positive x

• Function approaches zero or positive infinity for negative x

Calculating Inputs (x) and Outputs (y)

To calculate the outputs (y) of an exponential function when given an input (x), we simply substitute the value of x into the function expression and solve. For example, if the function is f(x) = 2 * 3^x and we want to find f(2), we substitute x for 2 to get f(2) = 2 * 3^2, resulting in f(2) = 18.

To determine the inputs (x) for a given output value (y), logarithms come into play. For instance, to solve the equation 4 * (1/2)^x = 1 for x, we divide both sides by 4 which leads to (1/2)^x = 1/4. Rewriting 1/4 as (1/2)^2, we equate the exponents, having x = 2.

Utilising logarithms is a highly effective method for solving exponential equations, particularly when the x values are not whole numbers. This technique allows us to manipulate the properties of exponential functions to isolate the needed variable and arrive at accurate solutions.

• Direct substitution to find outputs (y)

• Use of logarithms to find inputs (x)

• Solving exponential equations

Key Terms

• Exponential Function: A function of the form f(x) = a * b^x, where 'a' is a non-zero coefficient, 'b' is the base (b > 0 and b ≠ 1), and 'x' is the exponent.

• Coefficient: The value 'a' in an exponential function, which multiplies the base raised to the exponent.

• Base: The value 'b' in an exponential function, which is raised to the power of x and must be a positive constant different from 1.

• Exponent: The variable 'x' in an exponential function, indicating the power to which the base is raised.

• Exponential Growth: A characteristic of an exponential function where the base is greater than 1, resulting in a rapid increase in output as x increases.

• Exponential Decay: A characteristic of an exponential function where the base is between 0 and 1, resulting in a rapid decrease in output as x increases.

• Logarithm: A mathematical operation that is the inverse of exponentiation, used to solve exponential equations.

Important Conclusions

In this lesson, we explored the definition and characteristics of exponential functions, learning to identify their general structure f(x) = a * b^x, where 'a' is a non-zero coefficient and 'b' is a positive base different from 1. We discussed how these functions model phenomena involving exponential growth and decay, such as population growth and radioactive decay. We also examined the graphs of these functions and how they reflect exponential behaviour in various scenarios.

We covered how to calculate outputs (y) from inputs (x) and vice-versa, using direct substitution and logarithms to solve exponential equations. This not only enriched our understanding of how to work with these functions but also enabled us to find specific values in practical situations. We looked at real-world examples, such as the growth of bacteria in a culture and solving exponential equations.

Mastering exponential functions is essential since they find application across a multitude of fields, including biology, finance, and economics. Understanding how to model and predict exponential behaviours allows us to make better-informed decisions and gain deeper insights into our surroundings. I encourage everyone to keep delving into this topic, as mastering exponential functions will unlock a greater understanding of many complex phenomena.

Study Tips

• Review the practical examples discussed in class and try to solve additional problems related to exponential functions. Practicing with various scenarios will help solidify your understanding.

• Make use of online resources such as videos and graphing tools to visualise the behaviour of exponential functions. Visual aids can simplify the learning process by providing clarity on how these functions grow or decline.

• Study logarithms in greater depth, as they are key tools in resolving exponential equations. A solid grasp of logarithms will significantly aid in manipulating and solving these functions.