91Ó°ÊÓ

Function simplification is an essential skill in mathematics, particularly when dealing with functions and their equivalence. It involves breaking down complex expressions into simpler forms, making them easier to compare and understand.

In our exercise, we simplify each function step-by-step to identify their equivalence. Here is a detailed look at the process:
- We start with the function in its original form, such as \( f(x) = 40(0.625)^x \).
- For each function, we use algebraic techniques to rewrite them as simpler expressions.

This means recognizing equivalent forms, such as knowing that \( 0.625 = \frac{5}{8} \), and applying exponent rules where necessary.

Once simplified, these expressions allow us to see that they are, in fact, the same function. This process not only helps in solving problems but also deepens our understanding of how functions work.

Exponential functions are a fundamental part of higher mathematics, used in numerous real-life applications like population growth, radioactive decay, and financial forecasting. An exponential function has the form \( f(x) = a(b)^x \), where \( a \) is a constant, and \( b \) is the base raised to the power \( x \).

In our exercise:
- \( f(x) = 40(0.625)^x \) is an example of an exponential function.
- The base, \( 0.625 \), determines the rate at which the function grows or decays.

Exponential functions can have different bases, which change the nature of their growth or decay. But, as we simplified earlier, different-looking exponential functions can be equivalent if they simplify to the same form.

This is because the base's transformation, combined with the exponent's properties, can lead to the same fundamental growth pattern.

Understanding the properties of exponents is vital for simplifying exponential functions. They follow certain rules that allow us to manipulate the expressions easily and correctly. Here are some key properties:

1. **Product of powers**: \( a^m \times a^n = a^{m+n} \)
2. **Power of a power**: \( (a^m)^n = a^{mn} \)
3. **Power of a product**: \( (ab)^m = a^m \times b^m \)
4. **Negative exponents**: \( a^{-m} = \frac{1}{a^m} \)
5. **Fractional exponents**: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)

In our context, let's consider the function \( h(x) = 40\left(\frac{8}{5}\right)^{-x} \).

We use the property \( \left(\frac{a}{b}\right)^{-x} = \left(\frac{b}{a}\right)^x \) to simplify it:
- Thus, \( h(x) = 40\left(\frac{5}{8}\right)^x \) becomes equivalent to the other functions.

By using these properties effectively, we can solve and simplify complex exponential expressions.