Problem 11 - Algebra Form and Function

Question

Are the two functions the same function? $$ f(t)=450+30 t, \text { and } g(p)=450+30 p $$

Step-by-Step Solution

Answer

Answer: Yes, the functions f(t) = 450 + 30t and g(p) = 450 + 30p are the same. The only difference between them is the variable used, which does not affect the function's outcome or overall properties.

Step 1: Identifying the expressions of the functions

Given the functions: $$ f(t) = 450 + 30t \quad \text{and} \quad g(p) = 450 + 30p $$ The expression for function f(t) is \(450 + 30t\), and the expression for function g(p) is \(450 + 30p\).

Step 2: Comparing the expressions

With the expressions \(450 + 30t\) for f(t) and \(450 + 30p\) for g(p), we can see that the only difference between them is the variable: t in f(t) and p in g(p).

Step 3: Understanding the use of variables in functions

Variables in functions are essentially placeholders that represent any input within the domain of the function. The variable doesn't affect the outcome of the function or the function's overall properties. In this case, both t and p are used as input variables, and there are no restrictions on their domains.

Step 4: Conclusion

Since the only difference between the expressions of f(t) and g(p) is the variable used as input, we can conclude that these two functions are indeed the same function. The variables t and p are just different notations, but the functions have the same properties.

Key Concepts

Variable Notation

In algebra, variable notation refers to the symbol or letter used to represent a number or input within a mathematical expression or function. It's important to understand that the choice of variable is arbitrary and doesn't change the function itself. Whether you use "t", "p", or any other letter, it serves as a placeholder for any value from the function's domain.

• Variables are essential for creating general expressions that can work with any number within a given range.
• They allow us to describe and communicate complex concepts in a simplified way.
• Sometimes in different contexts, you might encounter different variables representing the same concept.

In the function expressions provided, both "t" and "p" are employed to signify the input of the functions. Despite the variations in variable notation, the actual function remains unchanged: they calculate the same outcomes for corresponding inputs.

Function Expressions

Function expressions are algebraic descriptions that define the relationship between the input and output of a function. They generally have a clear structure that includes a constant and a variable component.

Take, for instance, the expressions given:

• For \(f(t) = 450 + 30t\) and \(g(p) = 450 + 30p\), each contains a constant part, which is 450, and a variable part, \(30t\) or \(30p\).
• The variable part shows how the output changes relative to the input.
• The constant part often represents a starting point or base value.

These expressions might look different because they use distinct variables, but the mathematical operations are identical. This means the functions share identical behaviors and outputs, reinforcing the idea that different notations of the variable don't change the function's fundamental nature.

Domain of a Function

The domain of a function is the set of all possible input values (often represented by a variable) that the function can accept. In the example functions \(f(t)\) and \(g(p)\), the domain encompasses all values \(t\) or \(p\) can take. As there are no constraints mentioned, their domains are likely all real numbers.

Understanding domains helps in identifying which values are valid for use in functions, ensuring that calculations remain meaningful and correct.

• A function can be defined for real numbers, integers, positive numbers, or other specific sets as its domain.
• If a function involves operations like division by zero or taking roots of negative numbers, it might limit the domain.

For \(f(t) = 450 + 30t\) and \(g(p) = 450 + 30p\), the lack of restrictions means the functions are versatile, affecting how they can model various real-world situations. Recognizing the domain aids in interpreting the flexibility and application of functions.