Understanding Exponential Functions Through Point-Based Solutions

Exponential functions are a critical aspect of mathematical modeling, often used to analyze phenomena such as population growth, radioactive decay, and compound interest. In this article, we will explore how to determine the exponential function that passes through given points, using examples and step-by-step solutions.

Solving for the Exponential Function through Point-Based Equations

The general form of an exponential function is f(x) a b^x. Given two points, we can set up a system of equations to solve for the constants a and b. Let's take the points (2,2) and (3,4) and find the exponential function that passes through these.

Step 1: Set Up the System of Equations

For the point (2,2):

2 a b^2 tag{1}

For the point (3,4):

4 a b^3 tag{2}

Step 2: Express Constants and Solve for b

To solve for a and b, we start by expressing a from equation (1):

a frac{2}{b^2}

Substitute this expression for a into equation (2):

4 left(frac{2}{b^2}right) b^3

This simplifies to:

4 frac{2b^3}{b^2} 2b

Solve for b:

b frac{4}{2} 2

Step 3: Find Constant a

Now that we have b, we substitute it back into the expression for a:

a frac{2}{2^2} frac{2}{4} frac{1}{2}

Therefore, the exponential function is:

f(x) frac{1}{2} cdot 2^x

This can be simplified as:

f(x) 2^{x-1}

Exploring Another Form of Exponential Function

Another form of exponential function is f(x) a e^{b.x}, where a and b are constants. If we have the points (2,2) and (3,4), we set up the equations as:

2 a e^{2b} tag{1}

4 a e^{3b} tag{2}

To find b, we divide equation (2) by equation (1):

frac{4}{2} frac{a e^{3b}}{a e^{2b}} e^b

This simplifies to:

2 e^b

Solving for b gives:

b ln 2

Substitute b back into equation (1) to find a:

2 a e^{2ln 2} a e^{ln 2^2} 4a

This gives:

a frac{1}{2}

Therefore, the required exponential function is:

f(x) frac{1}{2} e^{xln 2} 2^{x-1}

Conclusion

By understanding how to solve for the constants in exponential functions, you can accurately model and predict various phenomena. Whether using the form (f(x) a b^x) or (f(x) a e^{b.x}), the key is to set up and solve the system of equations formed by the given points. This method can be applied in a wide range of fields from finance to biology.