Given F(x)=3x22 And G(x)=71/2x2, Find The Following Expressions.(a)(fg)(4)(b)(gf)(2)(c)(ff)(1)(d)(gg)(0)
Understanding how to evaluate composite functions is fundamental in algebra and calculus. In this article, we will explore the process of finding the values of composite functions such as (fg)(x), (gf)(x), (ff)(x), and (gg)(x) at specific points. Using the given functions F(x) = 3x²2 and G(x) = 71/2x², we will methodically compute each expression step by step, ensuring clarity and thorough understanding. Whether you're a student preparing for exams or a math enthusiast looking to strengthen your skills, this comprehensive guide will help demystify the process of evaluating composite functions.
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Understanding the Given Functions
Before diving into the calculations, it's crucial to interpret the given functions correctly. The functions provided are:
- F(x) = 3x22
- G(x) = 71/2x2
However, these expressions seem ambiguous at first glance. Let's clarify their likely intended forms based on standard notation and context:
Clarification of F(x)
The expression 3x22 probably means 3 × x × 2 × 2, which simplifies to:
- F(x) = 3 × x × 2 × 2 = 3 × x × 4 = 12x
Alternatively, it might have been intended as 3x²2, meaning 3 × x² × 2, which simplifies to:
- F(x) = 3 × x² × 2 = 6x²
Given the context and the common pattern of functions, it's more plausible that F(x) = 12x or F(x) = 6x². To ensure accuracy, let's analyze both possibilities and see which fits better with G(x) and the problem statements.
Clarification of G(x)
Similarly, 71/2x2 likely reads as (71/2) × x × 2 or (71/2) × x². Let's examine both:
- Option 1: G(x) = (71/2) × x × 2 = 71 × x
- Option 2: G(x) = (71/2) × x²
Given the structure, the first seems more straightforward, resulting in G(x) = 71x.
Final assumption for functions:
Based on the above, the most consistent and mathematically reasonable interpretation is:
- F(x) = 12x
- G(x) = 71x
This interpretation aligns with typical function forms and makes subsequent calculations straightforward.
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Revised Functions for Calculations
- F(x) = 12x
- G(x) = 71x
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Calculating Composite Functions
The problem requires evaluating the following:
- (a) (fg)(4)
- (b) (gf)(2)
- (c) (ff)(1)
- (d) (gg)(0)
Let's understand what each notation means:
- (fg)(x): Apply G to x, then apply F to the result.
- (gf)(x): Apply F to x, then apply G to the result.
- (ff)(x): Apply F twice, starting from x.
- (gg)(x): Apply G twice, starting from x.
Step-by-step calculations:
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(a) Find (fg)(4)
Step 1: Compute G(4)
\[
G(4) = 71 \times 4 = 284
\]
Step 2: Compute F(G(4)) = F(284)
\[
F(284) = 12 \times 284 = 3408
\]
Answer:
\[
(fg)(4) = 3408
\]
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(b) Find (gf)(2)
Step 1: Compute F(2)
\[
F(2) = 12 \times 2 = 24
\]
Step 2: Compute G(F(2)) = G(24)
\[
G(24) = 71 \times 24 = 1704
\]
Answer:
\[
(gf)(2) = 1704
\]
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(c) Find (ff)(1)
Step 1: Compute F(1)
\[
F(1) = 12 \times 1 = 12
\]
Step 2: Compute F(F(1)) = F(12)
\[
F(12) = 12 \times 12 = 144
\]
Answer:
\[
(ff)(1) = 144
\]
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(d) Find (gg)(0)
Step 1: Compute G(0)
\[
G(0) = 71 \times 0 = 0
\]
Step 2: Compute G(G(0)) = G(0)
\[
G(0) = 0
\]
Answer:
\[
(gg)(0) = 0
\]
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Summary of Results
| Expression | Calculation | Result |
|---------------------|--------------------------------|----------|
| (fg)(4) | F(G(4)) = 12 × 284 | 3408 |
| (gf)(2) | G(F(2)) = 71 × 24 | 1704|
| (ff)(1) | F(F(1)) = 12 × 12 | 144 |
| (gg)(0) | G(G(0)) = 71 × 0 | 0 |
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Conclusion and Key Takeaways
Evaluating composite functions involves carefully following the order of operations: first applying the inner function, then the outer function. The key steps include:
- Substituting the input value into the inner function.
- Calculating the result.
- Using the result as the new input for the outer function.
- Performing the outer function's calculation to obtain the final value.
In our problem, understanding the functions' forms was essential. Assuming F(x) = 12x and G(x) = 71x provided a straightforward path to the solutions. Always verify the function expressions carefully, especially when they are ambiguous or poorly formatted.
Additional Tips for Evaluating Composite Functions
- Break down the problem: Start from the innermost function.
- Substitute accurately: Replace variables with the given input values.
- Follow the order: Remember that (fg)(x) means F(G(x)), not G(F(x)), and vice versa.
- Check your calculations: Simple arithmetic errors can lead to incorrect results—double-check each step.
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Further Practice
To master composite functions, practice with various functions and different points. Try evaluating:
- Different combinations like (fg)(x), (gf)(x), (ff)(x), and (gg)(x) with other functions.
- More complex functions involving quadratic or exponential expressions.
- Inverse functions and their compositions.
Understanding these concepts not only prepares you for exams but also builds a strong foundation for calculus topics such as derivatives and integrals.
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Final Remarks
The process of evaluating composite functions is a fundamental skill in mathematics. By carefully interpreting the functions, substituting values, and following the correct order of operations, you can solve these problems efficiently. Remember, clarity in function notation and systematic calculations are your best tools for success.
If you keep practicing, you'll find that working with composite functions becomes intuitive, enabling you to tackle more advanced math topics with confidence.
