Problem 15 - Algebra Form and Function

Question

Are the two functions the same function? $$ Q(t)=\frac{t}{2}-\frac{3}{2} \text { and } P(t)=t-3 $$

Step-by-Step Solution

Answer

Explain your answer. Answer: No, the functions Q(t) and P(t) are not the same function. When comparing their expressions algebraically, Q(t) = (t/2) - (3/2) does not simplify to t - 3, which is P(t). Therefore, they are different functions.

Step 1: Rewrite the second function to have the same denominator

Since we are going to work with algebraic expressions, we should first rewrite the second function with the same denominator as the first one, which is 2: $$ P(t) = t - 3 = \frac{2t}{2} - \frac{6}{2} $$

Step 2: Compare the functions

Compare the expressions of the two functions: $$ Q(t) = \frac{t}{2} - \frac{3}{2} \;\;\;\; \text{and} \;\;\;\; P(t) = \frac{2t}{2} - \frac{6}{2} $$

Step 3: Check for equivalency

Test if these expressions are equal by simplifying and comparing: $$ Q(t): \frac{t}{2} - \frac{3}{2} \;\;\;\; P(t): \frac{2t - 6}{2} $$ Simplify $$P(t)$$: $$ P(t) = \frac{2(t - 3)}{2} = t - 3 $$

Step 4: Conclusion

Since the simplified expressions are not the same: $$ Q(t) = \frac{t}{2} - \frac{3}{2} \neq t - 3 = P(t) $$ The two functions Q(t) and P(t) are not the same function.

Key Concepts

Rational Expressions

Rational expressions are divisions of two polynomials. Think of them as fractions but with variables. In math, fractions have a numerator and a denominator; similarly, rational expressions are written as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)eq 0\).
These expressions appear often in algebra because they help us divide polynomials. We encounter rational expressions frequently in solving equations and simplifying complex problems. They're versatile tools that allow us to work through complicated calculations by breaking them down into simpler steps.
In our exercise, the first given function \(Q(t) = \frac{t}{2} - \frac{3}{2}\) is already in the form of a rational expression, where the whole expression has a common denominator of 2. This sets the stage for making it easier to compare to another function that might not be in fractional form!

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. It is a process where we make expressions easier to understand and work with by combining like terms and reducing fractions if possible.
When simplifying, you want to follow these steps:

• Identify like terms: Terms that have the same variables raised to the same power.
• Combine like terms: Add or subtract their coefficients.
• Reduce fractions: Simplify any fractional parts of the expression.
• Factor expressions: Look out for common factors that can be extracted.

In our solution, we simplified \(P(t) = \frac{2(t - 3)}{2}\) to \(t - 3\) to better match its structure with \(Q(t)\). By simplifying, we affirm whether the expressions equate and help clarify the relationship between different forms of the same expression.

Comparing Functions

Comparing functions in algebra involves examining their forms to determine if they represent the same relationship. This process helps identify function equivalence, meaning two functions produce the same output for any input.
To compare functions effectively:

• Align their representations, preferably to have the same denominator for ease.
• Simplify each function to its simplest form.
• Check for identical terms on both sides of the equation.

In our task, we compared two functions, \(Q(t)\) and \(P(t)\). By rewriting \(P(t)\) with the same denominator, we could directly compare the forms. After simplification, we saw that \(Q(t)\) and \(P(t)\) did not resolve to the same simplest form, meaning they are not equivalent functions.
This comparison technique helps us simplify and understand whether different representations capture the same mathematical relationship. Frequent practice with these comparisons enhances our ability to identify relationships and navigate complicated algebraic equations effectively.