Find conditions on a and b so that (f o g)(x) = x and (g o f)(x) = x for a Straight Line g(x)=ax + b

TL;DR

Finding conditions on a and b for commutativity of function compositions.

Transcript

hi everyone in this problem we have two functions f of x equals 3x minus 4 and g of x equals ax plus b and the question is to find conditions on a and b so that the function compositions actually commute kind of an interesting problem i haven't done it yet so this will be kind of interesting so solution so i'm thinking the way to do this is we just... Read More

Key Insights

• ⚾ Understanding function compositions involves substituting one function into another based on their definitions.
• 😃 Commutativity of function compositions requires the compositions f o g(x) and g o f(x) to be equal.
• 🍉 Solving for unknown coefficients in functions involves equating terms and matching coefficients.
• 😃 The derived condition b = 2 - 2a provides a clear relationship for the functions to commute.
• 😑 Mathematical problem-solving involves systematic calculations and manipulation of expressions.
• ❓ Properly distributing coefficients in function compositions ensures accurate results.
• 😫 Setting up equations based on function definitions is a crucial step in solving algebraic problems.

Questions & Answers

Q: What are the two functions given in the problem?

The two functions are f(x) = 3x - 4 and g(x) = ax + b, where a and b are coefficients to be determined for compositions to commute.

Q: How do you calculate f o g(x)?

To compute f o g(x), substitute g(x) into f(x) as f(ax + b) and simplify by distributing the coefficients to obtain the composition.

Q: What is the process to determine the conditions for function compositions to commute?

By setting f o g(x) equal to g o f(x) and comparing coefficients, a condition can be derived as b = 2 - 2a for the compositions to commute.

Q: How does the condition b = 2 - 2a ensure commutativity of function compositions?

The derived condition indicates that for the functions f and g to commute, the coefficient b must be 2 units greater than twice the coefficient a.

Summary & Key Takeaways

• Given two functions, f(x) = 3x - 4 and g(x) = ax + b, find conditions on a and b for their compositions to commute.

• Calculate f o g(x) and g o f(x) by substituting the inner function and applying function definitions.

• Equate the compositions and solve for a and b to obtain the condition b = 2 - 2a.