An interesting function in mathematics
The exponential function is a concept that sounds extremely difficult to get your head around. What exactly is an exponential function Fortunately, the explanation of this concept is much easier than you would imagine. The exponential function has the very simple formula: f(x) = a^x, with the ‘a’ having to be greater than zero in order to satisfy the condition that a > 0.
The name – intriguing and perhaps a little daunting to a rookie student of maths – comes from the fact that ‘x’ is in the exponent.
It is interesting to note that the graph of our function is such a curve that, no matter what, it always crosses the y-axis at point 1. However, depending on whether ‘a’ is greater than 1 or less than 1, it can look inverted.
The domain of any exponential function is the set of real numbers R – and its set of values are all positive and real numbers.
The graph that we get from an exponential function is called an exponential curve.
The exponential function has a number of domains or properties. Firstly, its domain is the set of real numbers. Secondly, only positive numbers belong to the set of values. Thirdly, the function can be either decreasing or increasing. In general, a function is differentiable. The value for the zero argument is 1. The graph IS NOT a straight line.
Contrary to popular belief, mathematics is not a difficult subject and can be learned. It is best when you practise because, as they say, practice makes perfect. It is worth looking for interesting tasks with solutions on the internet to practise your ability to calculate the exponential function. After a few such examples, you will find that you have mastered the material perfectly – and it is one of those things that can come in handy for your studies if you ever considering following the same career path as that of your teacher.
If you're still having trouble understanding what an exponential function is, it's worth reading this text a few times – the examples of graphs will help you better understand the issue and solve any tasks.
Key takeaway
Exponential function: Characterising the exponential function Read the definition of the exponential function and learn the best ways to understand it. What kind of formula do we use
When to use it?
Write the definition or relation behind "Exponential function".
Answer check
- Write the definition or relation behind "Exponential function".
- Use a simple numerical example and show every operation.
- Check whether the result satisfies the conditions of the task.
User-focused answer
Exponential function: Characterising the exponential function Read the definition of the exponential function and learn the best ways to understand it. What kind of formula do we use Use this topic when the task asks for more than a name: you must identify the condition, choose the rule and justify the result.
When this topic is actually needed
Use this topic when the task asks for more than a name: you must identify the condition, choose the rule and justify the result. Exponential function: definition, notation and example. Start with one clear definition sentence, then show the rule, and only then substitute the data.
The most common mistake is remembering the term but ignoring the condition in the task. If the answer has a unit, keep the unit with every number; if it is a language or glossary topic, show the term in a full sentence.
Complete way to work with the topic
- name the given data and the unknown
- write the definition or relationship
- test it on a simple example
- check the unit, range or sentence meaning
If the answer has a unit, keep the unit with every number; if it is a language or glossary topic, show the term in a full sentence. Start with one clear definition sentence, then show the rule, and only then substitute the data. If the answer has a unit, keep the unit with every number; if it is a language or glossary topic, show the term in a full sentence.
Worked example with commentary
Exponential function: Start with one clear definition sentence, then show the rule, and only then substitute the data. If the answer has a unit, keep the unit with every number; if it is a language or glossary topic, show the term in a full sentence.
| User-focused answer | What to remember |
|---|---|
| When this topic is actually needed | definition, notation and example |
| Mistakes that usually weaken the answer | The most common mistake is remembering the term but ignoring the condition in the task. |
| Complete way to work with the topic | If the answer has a unit, keep the unit with every number; if it is a language or glossary topic, show the term in a full sentence. |
Mistakes that usually weaken the answer
The most common mistake is remembering the term but ignoring the condition in the task. The most common mistake is remembering the term but ignoring the condition in the task. Start with one clear definition sentence, then show the rule, and only then substitute the data.
Explain the topic in your own words. Create an example that shows when the rule can be used. Name one possible mistake and correct it.
Check exercises
- Explain the topic in your own words.
- Create an example that shows when the rule can be used.
- Name one possible mistake and correct it.
What to remember: Exponential function. Use this topic when the task asks for more than a name: you must identify the condition, choose the rule and justify the result. Start with one clear definition sentence, then show the rule, and only then substitute the data.