Suppose That F(x)= (3x3) And G(x)=6x^28.For Each Function H Given Below, Find A Formula For H(x) And
Understanding how to manipulate and combine functions is a fundamental aspect of algebra and calculus. In this article, we will explore the process of determining explicit formulas for composite functions based on given functions F(x) and G(x). Specifically, when F(x) = 3x^3 and G(x) = 6x^28, we will examine various ways to form new functions H(x), such as sums, differences, products, quotients, and compositions. Our goal is to provide clear, step-by-step instructions to find formulas for each H(x) based on the operations involved, and to deepen your understanding of function operations for improved problem-solving skills and SEO-friendly content.
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Understanding the Given Functions and Basic Operations
Given Functions
- F(x) = 3x^3
- G(x) = 6x^28
These two functions are polynomial functions involving powers of x. F(x) is a cubic function scaled by 3, while G(x) involves a high-degree polynomial scaled by 6. When working with these functions, common operations include addition, subtraction, multiplication, division, and composition, each leading to a different form of H(x).
Function Operations and Their Significance
- Sum of functions: H(x) = F(x) + G(x)
- Difference of functions: H(x) = F(x) - G(x)
- Product of functions: H(x) = F(x) G(x)
- Quotient of functions: H(x) = F(x) / G(x), provided G(x) ≠ 0
- Composition of functions: H(x) = F(G(x)) or G(F(x))
Each of these operations results in a new function H(x) with a unique formula derived from the original functions. Let’s explore each case in detail.
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Formulas for Sum and Difference of F(x) and G(x)
Sum of F(x) and G(x): H(x) = F(x) + G(x)
To find this, simply add the expressions for F(x) and G(x):Calculation:
- F(x) = 3x^3
- G(x) = 6x^28
- H(x) = 3x^3 + 6x^28
Final Formula:
H(x) = 3x^3 + 6x^28
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Difference of F(x) and G(x): H(x) = F(x) - G(x)
Similarly, subtract G(x) from F(x):Calculation:
- F(x) = 3x^3
- G(x) = 6x^28
- H(x) = 3x^3 - 6x^28
Final Formula:
H(x) = 3x^3 - 6x^28
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Formulas for Product and Quotient of F(x) and G(x)
Product: H(x) = F(x) G(x)
Multiplying the two functions:Calculation:
- F(x) = 3x^3
- G(x) = 6x^28
- H(x) = (3x^3) (6x^28)
Simplification Steps:
- Multiply the coefficients: 3 6 = 18
- Add the exponents of x: 3 + 28 = 31
Final Formula:
H(x) = 18x^31
Quotient: H(x) = F(x) / G(x), G(x) ≠ 0
Dividing F(x) by G(x):Calculation:
- F(x) = 3x^3
- G(x) = 6x^28
- H(x) = (3x^3) / (6x^28)
Simplification Steps:
- Divide the coefficients: 3 / 6 = 1/2
- Subtract the exponents of x: 3 - 28 = -25
Final Formula:
H(x) = (1/2) x^{-25} = \frac{1}{2} \cdot \frac{1}{x^{25}}
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Formulas for Composition of Functions
Function composition involves plugging one function into another. There are two primary compositions to consider:
1. H(x) = F(G(x))
This means replacing every x in F(x) with G(x):Calculation:
- F(x) = 3x^3
- G(x) = 6x^28
- H(x) = 3 [G(x)]^3
Step-by-step:
- First, find G(x): 6x^28
- Then, cube G(x): [6x^28]^3 = 6^3 x^{28 3} = 216 x^{84}
- Finally, multiply by 3: H(x) = 3 216 x^{84} = 648 x^{84}
Final Formula:
H(x) = 648x^{84}
2. H(x) = G(F(x))
Replace x in G(x) with F(x):Calculation:
- G(x) = 6x^28
- Replace x with F(x): 6 [F(x)]^{28}
Step-by-step:
- F(x) = 3x^3
- Calculate [F(x)]^{28} = (3x^3)^{28} = 3^{28} x^{3 28} = 3^{28} x^{84}
- Multiply by 6: H(x) = 6 3^{28} x^{84}
Final Formula:
H(x) = 6 3^{28} x^{84}
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Practical Applications and Tips for Working with Function Formulas
Why Understanding These Formulas Matters
Knowing how to derive formulas for combined functions allows mathematicians, engineers, and scientists to analyze complex systems more effectively. Whether modeling physical phenomena, analyzing data, or solving equations, understanding these operations is crucial.Tips for Simplifying Function Expressions
- Always simplify coefficients first.
- Use exponent rules: x^a x^b = x^{a + b}, (x^a)^b = x^{a b}, x^a / x^b = x^{a - b}.
- When composing functions, carefully substitute and simplify step-by-step.
- Check for domain restrictions, especially in division and composition, to avoid undefined expressions.
Common Mistakes to Avoid
- Forgetting to raise coefficients to the power when exponentiating products.
- Mixing up addition and multiplication rules for exponents.
- Neglecting to consider the domain restrictions in division and composition.
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