Summary
Highlights
This section combines rational and square root functions, emphasizing that the expression under the radical in the denominator must be strictly greater than zero (cannot be zero). This affects both vertical asymptotes and range calculations.
The video begins by defining domain as all possible x-values and range as all possible y-values of a function. It illustrates with an example graph, explaining how to identify the lowest and highest x and y values, and the use of parentheses (open circle) and brackets (closed circle) in interval notation.
Further examples of finding domain and range from more complex graphs are presented, including those with arrows indicating continuation to infinity and breaks or gaps in the function, requiring the use of union notation.
The video demonstrates how to find domain and range for functions with vertical and horizontal asymptotes. It emphasizes that the function never touches these asymptotes, leading to their exclusion from the domain (vertical asymptote) and range (horizontal asymptote).
For linear equations, the domain and range are always all real numbers, represented as negative infinity to positive infinity, as there are no restrictions on x or y values.
The domain of all polynomial functions, including quadratic functions, is all real numbers. The video explains how to find the range by identifying the vertex (minimum or maximum y-value) of the parabola, using methods like factoring or the vertex formula (x = -b/2a).
For cubic functions, the domain and range are also always all real numbers. The video illustrates how transformations (shifts) affect the graph but not the overall domain and range for these types of polynomials.
When dealing with square root functions (even index radical), the expression inside the radical must be greater than or equal to zero. The video explains how reflections over the x and y axes influence the domain and range.
Examples showcase how to determine the starting point and direction of the graph (which quadrant it extends into) based on transformations. This helps in identifying the domain and range, which are restricted by the values inside and outside the radical.
The video explains how to find the domain for square root functions containing quadratic expressions by factoring the quadratic and using a number line to test intervals where the expression is non-negative. The range is determined by whether there's a positive or negative sign in front of the radical.
For rational functions, the domain excludes values that make the denominator zero (vertical asymptotes). The range excludes the horizontal asymptote, which is determined by comparing the degrees of the numerator and denominator.
A detailed example of a rational function with both a vertical asymptote and a 'hole' in the graph (due to canceling factors) is provided. This demonstrates how both must be excluded from the domain, and how the hole's y-coordinate is excluded from the range alongside the horizontal asymptote.
The domain for absolute value functions is always all real numbers. The range is determined by the vertex and whether the graph opens upward or downward, which is influenced by a negative sign in front of the absolute value expression.
An explanation of exponential functions shows that their domain is all real numbers. The range is restricted by a horizontal asymptote, which is determined by the constant added or subtracted from the exponential term.
The domain of logarithmic functions is restricted: the argument of the logarithm must be strictly greater than zero, which also defines a vertical asymptote. The range for logarithmic functions is always all real numbers.
This segment provides a concise summary, contrasting the domain and range characteristics of exponential and logarithmic functions, highlighting their respective asymptotes and growth patterns.
For sine and cosine functions, the domain is always all real numbers. The range is determined by the amplitude and any vertical shifts, typically varying between -1 and 1, but can change.
Tangent functions have an unlimited range (all real numbers) but a restricted domain due to vertical asymptotes at odd multiples of pi/2. These asymptotes must be excluded from the domain.