Determine whether each function graphed or defined is one-to-one. y = x+4 / x-3

Verified step by step guidance

1. Understand that a function is one-to-one if each output value is paired with exactly one input value, meaning it passes the Horizontal Line Test.
2. Consider the function \( y = \frac{x+4}{x-3} \). This is a rational function, which can be analyzed for one-to-one properties.
3. To determine if the function is one-to-one, check if it passes the Horizontal Line Test. This involves ensuring that no horizontal line intersects the graph of the function more than once.
4. Alternatively, algebraically determine if the function is one-to-one by checking if \( f(a) = f(b) \) implies \( a = b \). Set \( \frac{a+4}{a-3} = \frac{b+4}{b-3} \) and solve for \( a = b \).
5. Simplify the equation \( (a+4)(b-3) = (b+4)(a-3) \) and see if it leads to \( a = b \) for all values in the domain of the function.

Key Concepts

One-to-One Function

A one-to-one function is a type of function where each output value is associated with exactly one input value. This means that no two different inputs produce the same output. To determine if a function is one-to-one, one can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Graphing Rational Functions

Rational functions are expressed as the ratio of two polynomials. The function given, y = (x + 4) / (x - 3), is a rational function. Understanding how to graph rational functions involves identifying asymptotes, intercepts, and the general shape of the graph, which can help in visualizing whether the function is one-to-one.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For the function y = (x + 4) / (x - 3), there is a vertical asymptote at x = 3, where the function is undefined. Recognizing asymptotes is crucial for understanding the behavior of the function and can influence whether the function is one-to-one, especially in the context of its range.