Problem 93 - Elementary and Intermediate Algebra

Question

Write out in words how to say each of the following: $$ (f \circ g)(2) \quad g(f(-8)) $$

Step-by-Step Solution

Answer

(f ∘ g)(2) is "f of g of 2" and g(f(-8)) is "g of f of negative 8."

Step 1: Understanding the Function Notation

The notation \((f \circ g)(x)\) means "the composition of function \(f\) with function \(g\) at \(x\)". For the first expression, we replace \(x\) with 2.

Step 2: Breaking Down the First Expression

For \((f \circ g)(2)\), say it as "f of g of 2." This means you take 2, apply the function \(g\) to find \(g(2)\), then apply \(f\) to the result \(f(g(2))\).

Step 3: Understanding the Second Composition Notation

The expression \(g(f(-8))\) reads as "g of f of negative 8." Here, you first find \(f(-8)\), then use the result as the input to \(g\).

Step 4: Breaking Down the Second Expression

Write \(g(f(-8))\) as "g of f of negative 8." This indicates applying \(f\) to \(-8\), then applying \(g\) to that result.

Key Concepts

Composition of Functions

Composition of functions is a fundamental concept in mathematics that allows you to combine two functions into one. It is symbolized by the circle notation: \((f \circ g)(x)\). In simpler terms, it means applying two functions in sequence: first, you apply the second function, \(g\), to an input \(x\), and then you apply the first function, \(f\), to the result of \(g(x)\). For example, to find \((f \circ g)(2)\), you would first evaluate \(g(2)\), and then use this output as the input for \(f\). This process can be expressed in words as "f of g of 2." When interpreting composition of functions, remember:

• The order of functions matters—always start from the inside and move out.
• Write out or verbalize each step to clearly follow the flow of operations.
• Verify each calculated result, as each relies on the prior computation.

This concept is particularly useful in calculus and complex algebra, where combining functions simplifies solving equations and modeling relationships.

Mathematical Expressions

Mathematical expressions are structured arrangements of numbers, variables, and operation symbols that represent specific values. In the context of function notation, expressions such as \((f \circ g)(2)\) and \(g(f(-8))\) tell us not only which numbers or variables to use, but also the sequence of operations to perform.
When handling mathematical expressions related to functions:

• Identify each function and its respective input clearly.
• Keep track of the order of operations, especially when multiple functions are involved.
• Bracket each function operation to maintain clarity in sequence.

The suggestion to verbalize expressions, as with "f of g of 2" or "g of f of negative 8," helps ensure that each step in solving is logical and follows correctly from the last step. This builds a mental model that helps in understanding and communicating mathematical processes effectively.

Function Operations

Function operations involve different processes that can be performed on functions, such as addition, subtraction, multiplication, division, and composition. Composition is one type of operation, where two functions are applied in succession. However, other function operations modify or combine the output values of two or more functions directly.
In function operations:

• Function addition means you add the outputs: \( (f+g)(x) = f(x) + g(x) \).
• With subtraction, subtract the outputs: \( (f-g)(x) = f(x) - g(x) \).
• Multiplication involves multiplying outputs: \( (f \cdot g)(x) = f(x) \cdot g(x) \).
• For division, divide the outputs: \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \), provided \(g(x) eq 0\).

Understanding these operations extends your ability to manipulate functions algebraically and apply them in broader mathematical contexts. When dealing with composition specifically, as in our example with \((f \circ g)(2)\) or \(g(f(-8))\), always be diligent in applying functions in the correct order, as the resulting values depend on it.