To determine if the statement "The domain of a function is the set of all outputs. Many times, it is the set of all y-values." is true or false, let's break down the concepts involved:
1. Domain of a Function:
- The domain of a function is the set of all possible input values [tex]\( x \)[/tex] for which the function is defined. In simpler terms, it's every [tex]\( x \)[/tex]-value that you can put into the function and get a valid output.
2. Outputs of a Function:
- The outputs of a function are the values that the function produces. These are dependent on the input values [tex]\( x \)[/tex] and are typically represented by [tex]\( y \)[/tex] or [tex]\( f(x) \)[/tex].
3. Range of a Function:
- The range of a function is the set of all possible output values [tex]\( y \)[/tex]. These are the values that the function can produce given all possible inputs.
Now, let’s examine the statement:
"The domain of a function is the set of all outputs. Many times, it is the set of all y-values."
- The first part claims that the domain is the set of all outputs. This is incorrect. The domain is the set of all inputs, not outputs.
- The second part mentions that it is the set of all y-values. However, the y-values correspond to the range, not the domain.
Putting all of this together, the correct understanding is:
- The domain is related to inputs (x-values).
- The range is related to outputs (y-values).
Based on this understanding, the statement "The domain of a function is the set of all outputs. Many times, it is the set of all y-values." is false.
