91Ó°ÊÓ

In mathematics, functions are a fundamental concept that represents a relationship between a set of inputs and outputs. They define how each element from a "domain" (input set) is related to an element in the "range" (output set). Functions are often expressed as equations such as \( f(x) = x + 1 \), where \( f \) is the function, \( x \) is the variable or input, and \( x + 1 \) is the rule that transforms the input into the output.
When analyzing functions, especially in compositions, it’s crucial to understand the 'order of operations'. Function composition is when one function provides input to another function, such as \((f \circ g)(x)\), meaning \( f(g(x)) \). The key to function composition is substituting the output of one function directly into another function.
In our example: \( f(x)=x+1 \) and \( g(x)=6x \), these are both simple linear functions. To find \( (f \circ g)(x) \), we must substitute \( g(x) \) into \( f(x) \) to find the composite result.

The concepts of domain and range help describe and understand functions better. Here, the domain is the set of all possible input values for which the function is defined, while the range is the set of all possible outputs.
For the function \( f(x) = x + 1 \):

• Domain: All real numbers, since you can add 1 to any real number.
• Range: All real numbers, because there is no restriction on the values of \( x + 1 \).

Similarly, for \( g(x) = 6x \):

• Domain: All real numbers, because multiplying any real number by 6 gives another real number.
• Range: All real numbers, following the same logic.

When we apply function composition, such as \((f \circ g)(x)\) and \((g \circ f)(x)\), the resulting domain and range might not immediately change, yet understanding these concepts ensures you are aware of all possible function outputs.

Mathematical proofs are logical arguments that confirm whether statements or propositions are true or false. In this context, we were asked to verify the statement \((f \circ g)(x)=(g \circ f)(x)\). To do so, we perform calculations to compare the results of \( f(g(x)) \) and \( g(f(x)) \), breaking the process into understandable steps.
In our example:

• Step 1: Compute \( f(g(x)) \) and find it equals \( 6x+1 \).
• Step 2: Compute \( g(f(x)) \) and see it equals \( 6x+6 \).
• Step 3: Compare results to determine that \( 6x+1 eq 6x+6 \), proving the initial statement is false.

The importance of proof in mathematics is to ensure clarity and correctness of results. By establishing that \((f \circ g)(x)\) is not equal to \((g \circ f)(x)\), we confirm that function composition is not commutative, meaning \( f \circ g \) does not generally equal \( g \circ f \). This highlights the significance of logical reasoning and systematic computation in mathematical proofs.