Domain and Range of Combined Functions
Domain and range of combined functions is a Grade 11 Algebra 1 topic from enVision Chapter 10 that teaches students how to find valid inputs and outputs when two functions are added, subtracted, or multiplied. The domain of (f ± g)(x) or (f · g)(x) is the intersection of the individual domains — values that work for both functions simultaneously. For example, if f(x) = √x (domain x ≥ 0) and g(x) = 1/(x-2) (domain x ≠ 2), the combined domain is x > 2. Range must be analyzed from the combined expression itself, not from the individual ranges.
Key Concepts
For combined functions $(f \pm g)(x)$ or $(f \cdot g)(x)$, the domain is the intersection of the domains of $f(x)$ and $g(x)$. The range must be determined by analyzing the behavior of the combined function.
Common Questions
How do you find the domain of a combined function?
Find the intersection of the domains of both original functions. Any x-value that is excluded from either function must be excluded from the combined function.
If f(x) = √x and g(x) = 1/(x-2), what is the domain of (f + g)(x)?
The domain is x > 2. We need x ≥ 0 from f(x) and x ≠ 2 from g(x). The intersection of these restrictions gives x ≥ 0 and x ≠ 2, which simplifies to x > 2.
Can the range of (f + g)(x) be found by combining the ranges of f and g?
No. The range of a combined function must be determined by analyzing the behavior of the actual combined expression. Individual ranges do not simply combine.
What is the domain of (f · g)(x) where f(x) = x² and g(x) = 3?
All real numbers. f(x) = x² has domain all reals, and g(x) = 3 also has domain all reals, so their intersection is all real numbers.
Why might combining functions restrict the domain further?
Because you need the input to work for both functions at the same time. A value excluded from either function cannot be used in the combined function.
How does (f + g)(x) = x² + 3 demonstrate range analysis?
Even though f(x) = x² has range y ≥ 0 and g(x) = 3 has range y = 3, the combined function x² + 3 has range y ≥ 3, which must be derived from the combined expression.