Explain how to find the domain of fo g if you know the domain and range of f and Choose the correct answer below The domain of f o consists of all x in the domain of The domain of f 0 g consists of all x in the domain of f Identify the X-values in the domain of g whose corresponding range values are in the domain of f dentify the x-values in the domain of whose corresponding range values are in the domain of g-
The Correct Answer and Explanation is:
To find the domain of f∘gf \circ gf∘g, also written as (f∘g)(x)(f \circ g)(x)(f∘g)(x), where fff and ggg are two functions, we need to take into account both the domain and range of the individual functions. Here’s a breakdown of how to approach this:
Step 1: Understand the Domains and Ranges of fff and ggg
- Domain of fff: This is the set of all input values xxx for which f(x)f(x)f(x) is defined.
- Range of ggg: This is the set of all possible output values that g(x)g(x)g(x) can produce.
- Domain of ggg: This is the set of all input values xxx for which g(x)g(x)g(x) is defined.
- Range of fff: This is the set of all possible output values that f(x)f(x)f(x) can produce.
Step 2: Understand the Composition of Functions f∘gf \circ gf∘g
The composition f∘gf \circ gf∘g means that you first apply the function ggg to xxx, and then apply the function fff to the result of g(x)g(x)g(x). For the composition to be valid, the output of g(x)g(x)g(x) must lie within the domain of fff, because fff will only be able to process values that are in its domain.
Step 3: Identify the Domain of f∘gf \circ gf∘g
To determine the domain of f∘gf \circ gf∘g, follow these two conditions:
- xxx must be in the domain of ggg: This is because we need to be able to input xxx into the function ggg.
- g(x)g(x)g(x) must be in the domain of fff: After applying g(x)g(x)g(x), the result must be a value that fff can handle, meaning that g(x)g(x)g(x) must lie in the domain of fff.
So, the domain of f∘gf \circ gf∘g consists of all xxx-values that are in the domain of ggg, but for which g(x)g(x)g(x) lies within the domain of fff.
Correct Answer:
- The domain of f∘gf \circ gf∘g consists of all xxx in the domain of ggg such that g(x)g(x)g(x) is in the domain of fff.
Example:
Let’s consider an example with specific functions:
- Let f(x)=xf(x) = \sqrt{x}f(x)=x, where the domain of fff is x≥0x \geq 0x≥0 (because the square root function is only defined for non-negative numbers).
- Let g(x)=x2g(x) = x^2g(x)=x2, where the domain of ggg is all real numbers xxx.
To find the domain of f∘gf \circ gf∘g, we first identify the domain of ggg, which is all real numbers, and then check for which xxx-values the output g(x)=x2g(x) = x^2g(x)=x2 lies within the domain of fff. Since f(x)f(x)f(x) is defined for non-negative numbers, we need x2≥0x^2 \geq 0x2≥0, which is true for all xxx. Therefore, the domain of f∘gf \circ gf∘g is all real numbers.
