Functions


In Exercises 1–6, find the domain and range of each function.


f(t) = 4/(3 − t)

Verified step by step guidance

1. To find the domain of the function \( f(t) = \frac{4}{3 - t} \), we need to determine the values of \( t \) for which the function is defined. The function is undefined when the denominator is zero.
2. Set the denominator equal to zero and solve for \( t \): \( 3 - t = 0 \).
3. Solving \( 3 - t = 0 \) gives \( t = 3 \). Therefore, the function is undefined at \( t = 3 \).
4. The domain of \( f(t) \) is all real numbers except \( t = 3 \). In interval notation, this is \( (-\infty, 3) \cup (3, \infty) \).
5. The range of \( f(t) \) is all real numbers except zero, because the function can take any real value except zero as \( t \) approaches 3 from either side, making the function approach positive or negative infinity.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions like f(t) = 4/(3 - t), the domain excludes any values that make the denominator zero, as division by zero is undefined. In this case, t cannot equal 3, so the domain is all real numbers except t = 3.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function f(t) = 4/(3 - t), as t approaches 3, the function approaches infinity or negative infinity, depending on the direction of approach. Thus, the range includes all real numbers except for the value that the function approaches as t nears 3, which is 0.

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never touches or crosses, typically occurring where a function is undefined. In the case of f(t) = 4/(3 - t), the vertical asymptote is at t = 3, indicating that as t approaches 3 from either side, the function's value increases or decreases without bound. This concept is crucial for understanding the behavior of the function near its domain restrictions.