UECE-2021 Function Problem Solving F(x) = 8ax

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#h1 Introduction

This article provides a detailed solution and explanation for the UECE-2021 mathematics problem involving a real function of a real variable, f(x) = 8a^x. We will explore the concepts of exponential functions and algebraic manipulation to find the value of f(4) ÷ f(5). This article aims to help students and enthusiasts understand the problem-solving process, reinforcing key mathematical concepts and techniques. Let's delve into the step-by-step solution.

#h2 Understanding the Problem

Before we dive into the solution, let's make sure we fully understand the problem statement. We are given a function f(x) = 8a^x, where a is a positive real number not equal to 1. We are also given that f(3) = 125. The goal is to find the value of f(4) ÷ f(5). This involves understanding exponential functions, algebraic manipulation, and substitution. To break it down further, we need to first find the value of a using the given condition f(3) = 125. Once we have a, we can compute f(4) and f(5) and finally calculate their quotient. This requires a methodical approach, paying close attention to detail to avoid common pitfalls in algebraic manipulation.

#h2 Step-by-Step Solution

Step 1: Find the Value of a

We are given that f(3) = 125. Substitute x = 3 into the function definition:

f(3) = 8a^3 = 125

To find a, we need to isolate a^3:

a^3 = 125 / 8

Now, take the cube root of both sides:

a = ∛(125 / 8)

Since 125 = 5^3 and 8 = 2^3, we have:

a = ∛(5^3 / 2^3) = 5 / 2

So, a = 5/2. This value will be crucial in determining f(4) and f(5). Finding a correctly is the foundation for the rest of the solution. We will use this value to compute the function values at x = 4 and x = 5.

Step 2: Calculate f(4)

Now that we have a, we can find f(4) by substituting x = 4 and a = 5/2 into the function:

f(4) = 8(5/2)^4

Calculate (5/2)^4:

(5/2)^4 = 5^4 / 2^4 = 625 / 16

Now, multiply by 8:

f(4) = 8 * (625 / 16) = 625 / 2

Thus, f(4) = 625 / 2. This intermediate result is necessary for the final calculation. We will use this value along with f(5) to find the desired quotient.

Step 3: Calculate f(5)

Next, we find f(5) by substituting x = 5 and a = 5/2 into the function:

f(5) = 8(5/2)^5

Calculate (5/2)^5:

(5/2)^5 = 5^5 / 2^5 = 3125 / 32

Now, multiply by 8:

f(5) = 8 * (3125 / 32) = 3125 / 4

Thus, f(5) = 3125 / 4. This is another key result that we will use to find the final answer. With both f(4) and f(5) calculated, we are now ready to find their ratio.

Step 4: Calculate f(4) ÷ f(5)

We want to find the value of f(4) ÷ f(5). We have:

f(4) ÷ f(5) = (625 / 2) ÷ (3125 / 4)

To divide fractions, multiply by the reciprocal of the second fraction:

f(4) ÷ f(5) = (625 / 2) * (4 / 3125)

Simplify the fractions:

f(4) ÷ f(5) = (625 * 4) / (2 * 3125)

f(4) ÷ f(5) = 2500 / 6250

Reduce the fraction by dividing both numerator and denominator by their greatest common divisor, which is 2500:

f(4) ÷ f(5) = 2500 / 6250 = 2 / 5

Therefore, f(4) ÷ f(5) = 2 / 5. This is the final answer to the problem. We have successfully used the given information and algebraic manipulation to arrive at the solution.

#h2 Alternative Method Using Properties of Exponents

An alternative method to solve this problem involves using the properties of exponents, which can sometimes simplify the calculations. We know that f(x) = 8a^x, and we want to find f(4) / f(5). Instead of calculating f(4) and f(5) separately, we can use the properties of exponents to simplify the expression directly. This method is particularly useful when dealing with exponential functions as it reduces the number of individual calculations and the chances of making a mistake. Let's explore this approach.

Step 1: Express the Ratio Using the Function Definition

First, express the ratio f(4) / f(5) using the function definition:

f(4) / f(5) = (8a^4) / (8a^5)

Step 2: Simplify the Expression

Notice that the 8 in the numerator and denominator cancels out:

(8a^4) / (8a^5) = a^4 / a^5

Now, use the property of exponents a^m / a^n = a^(m-n):

a^4 / a^5 = a^(4-5) = a^(-1)

So, f(4) / f(5) = a^(-1) = 1 / a

This simplification significantly reduces the amount of calculation required. We only need to find the value of a and then take its reciprocal.

Step 3: Use the Value of a

From our previous calculation, we found that a = 5/2. Substitute this value into the simplified expression:

1 / a = 1 / (5/2)

Taking the reciprocal gives:

1 / (5/2) = 2 / 5

Therefore, using the properties of exponents, we directly found that f(4) / f(5) = 2 / 5, which matches our previous result. This method highlights the efficiency of using exponent rules in problem-solving.

#h2 Common Mistakes and How to Avoid Them

When solving problems involving exponential functions, several common mistakes can occur. Being aware of these pitfalls and how to avoid them can significantly improve your problem-solving accuracy. Here are some typical errors and strategies to prevent them:

  1. Incorrectly Calculating the Value of a:

    • Mistake: Making errors when taking the cube root or simplifying fractions.
    • Prevention: Double-check your calculations. Ensure you correctly identify the cube root and simplify fractions to their lowest terms. For example, when finding a from a^3 = 125 / 8, ensure you take the cube root of both the numerator and the denominator separately.

  2. Errors in Exponentiation:

    • Mistake: Miscalculating powers, especially when dealing with fractions.
    • Prevention: Practice calculating powers, especially those involving fractions. Remember that (a/b)^n = a^n / b^n. For instance, correctly compute (5/2)^4 as 5^4 / 2^4 = 625 / 16.

  3. Arithmetic Mistakes:

    • Mistake: Simple arithmetic errors when multiplying or dividing fractions.
    • Prevention: Write out each step clearly and double-check your arithmetic. Use a calculator for complex calculations, but still, review each step to ensure accuracy. For example, when dividing (625 / 2) by (3125 / 4), ensure you correctly multiply by the reciprocal.

  4. Forgetting to Simplify:

    • Mistake: Not simplifying the final fraction, leading to an unsimplified answer.
    • Prevention: Always simplify your final answer. Reduce fractions to their lowest terms. In our example, ensure you reduce 2500 / 6250 to 2 / 5.

  5. Misunderstanding Function Notation:

    • Mistake: Incorrectly substituting values into the function or misinterpreting what the question is asking.
    • Prevention: Ensure you understand function notation. f(3) = 125 means that when x = 3, the function value is 125. Carefully substitute values and re-read the question to ensure you are answering what is asked.

By being mindful of these common mistakes and implementing strategies to avoid them, you can increase your accuracy and confidence in solving similar problems.

#h2 Conclusion

In this article, we have thoroughly addressed the UECE-2021 mathematics problem involving the function f(x) = 8a^x. We successfully found the value of f(4) ÷ f(5) using both a step-by-step method and an alternative approach leveraging properties of exponents. The key steps included determining the value of a, calculating f(4) and f(5), and simplifying the final expression. Additionally, we discussed common mistakes to avoid when solving similar problems. Mastering these techniques enhances your problem-solving skills and provides a solid foundation for tackling more complex mathematical challenges. Remember, consistent practice and a clear understanding of fundamental concepts are essential for success in mathematics.