Physics and Mathematics
Exponential Function
1. Concept Overview
An exponential function is a function in which the variable appears in the exponent.
These functions are extremely important because they describe rapid growth or decay, such as:
- population growth
- radioactive decay
- compound interest
2. Definition of Exponential Function
A function [f] is called an exponential function if it is of the form
[f(x)=a^x]
where:
- [a] is a positive real constant
- [a≠1]
- is a real variable
3. Examples of Exponential Functions
- [f(x)=2^x]
- [g(x)=3^x]
- [h(x)=\dfrac{1}{2}^x]
4. Domain and Range
Domain:
All real numbers
[(−∞,∞)]
Range:
All positive real numbers
[(0,∞)]
Important:
An exponential function is never zero or negative.
5. Important Properties
For [f(x)=a^x]:
- [f(0)=1]
- [f(x)>0] for all [x]
- If [a>1], function is increasing
- If [0<a<1], function is decreasing
- Graph always passes through [(0,1)]
6. Increasing and Decreasing Nature
- If [a>1] → Exponential growth
- If [0<a<1] → Exponential decay
7. Graphical Behaviour
- The graph never touches the x-axis
- The x-axis is a horizontal asymptote
- Graph is always smooth and continuous
8. Conceptual Questions with Solutions
1. What is an exponential function?
A function where the variable is in the exponent is called an exponential function.
2. Why must base be positive?
Because negative bases do not give real values for all real x.
3. Why is a ≠1?
If [a=1], then [f(x)=1], which is a constant function, not exponential.
4. Is [f(x)=2^x] defined for all real x?
Yes. Its domain is all real numbers.
5. Can exponential function be zero?
No. Its range is only positive values.
6. What is the value of [a^0]?
For any valid base, [a^0=1].
7. Why is exponential function always positive?
Because a positive base raised to any real power remains positive.
8. Is exponential function one–one?
Yes. It is a strictly monotonic function.
9. Does exponential function have inverse?
Yes. Its inverse is a logarithmic function.
10. Why does graph not touch x-axis?
Because the function value is never zero.
11. Is exponential function continuous?
Yes. It is continuous for all real x.
12. Can base be a fraction?
Yes, if it lies between 0 and 1.
13. What happens when x is negative?
The function still gives a positive value.
14. Why is x-axis an asymptote?
Because the graph approaches but never touches the x-axis.
15. Where does graph always pass?
Through the point [(0,1)].
9. FAQ / Common Misconceptions
1. Exponential function can be negative.
False. It is always positive.
2. Base can be zero.
False. Base must be positive.
3. Domain is only positive x.
False. Domain is all real numbers.
4. [a^x] is polynomial.
False. Variable is in exponent.
5. Graph cuts x-axis.
False. It never touches x-axis.
6. [1^x] is exponential.
False. It gives a constant function.
7. Exponential functions are decreasing.
False. They can be increasing or decreasing.
8. Range includes zero.
False. Range is (0,∞).
9. Negative exponent gives negative value.
False. Value is still positive.
10. Exponential functions are not useful.
False. They model real-life growth and decay.
10. Practice Questions with Step by Step Solution
Question 1.
Find the value of [2^0].
Step-by-Step Solution:
- Any positive number raised to power zero is equal to [1].
Conclusion:
[2^0 = 1]
Question 2.
Evaluate [3^{−2}].
Step-by-Step Solution:
- Negative exponent means reciprocal.
- [3^{−2} = \dfrac{1}{3^2}].
- [3^2 = 9].
Conclusion:
[3^{−2} = \dfrac{1}{9}]
Question 3.
Find the domain of [f(x)=2^x].
Step-by-Step Solution:
- Exponential function is defined for all real values of [x].
Conclusion:
Domain = [(−∞,∞)]
Question 4.
Find the range of [f(x)=5^x].
Step-by-Step Solution:
- Exponential function is always positive.
- It never becomes zero or negative.
Conclusion:
Range = [(0,∞)]
Question 5.
Find the value of [f(0)] if [f(x)=7^x].
Step-by-Step Solution:
- Substitute [x=0].
- [7^0 = 1].
Conclusion:
[f(0)=1]
Question 6.
State whether the function [f(x)=\dfrac{1}{2}^x] is increasing or decreasing.
Step-by-Step Solution:
- Base [\dfrac{1}{2}] lies between [0] and [1].
- Exponential functions with base between [0] and [1] are decreasing.
Conclusion:
The function is decreasing.
Question 7.
State whether the function [f(x)=4^x] is one–one.
Step-by-Step Solution:
- Exponential functions are strictly monotonic.
- Strictly monotonic functions are one–one.
Conclusion:
The function is one–one.
Question 8.
Does the graph of [f(x)=3^x] intersect the x-axis?
Step-by-Step Solution:
- Exponential function is always positive.
- It never becomes zero.
Conclusion:
The graph does not intersect the x-axis.
Question 9.
Find [f(1)] and [f(2)] if [f(x)=2^x].
Step-by-Step Solution:
- Substitute [x=1]: [f(1)=2^1=2].
- Substitute [x=2]: [f(2)=2^2=4].
Conclusion:
[f(1)=2] and [f(2)=4]
Question 10.
Find the value of [a] if [a^0 = 1].
Step-by-Step Solution:
- For any positive number [a], [a^0=1].
Conclusion:
[a] can be any positive real number.
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