Given the precise function shown below, select all of the statements that are true. {(2x,x<1),(5,x=1):}
To determine which statements are true, we need to analyze the given function and the conditions for each interval:
1. {(2x, x<1):}
- This interval indicates that for any value of x that is less than 1, the function will output 2 times that value.
2. {(5, x=1):}
- This interval indicates that when x is equal to 1, the function will output a constant value of 5.
Based on this information, let's evaluate the given statements:
1. The function is defined for all real numbers.
- This statement is false because the given function is only defined for two specific intervals: x<1 and x=1.
2. The function is continuous for all real numbers.
- This statement is false because the function has two distinct intervals, which means it is not continuous over the entire real number line.
3. The function is discontinuous at x=1.
- This statement is true because the function jumps from a different output value (2x) to a constant value (5) when x=1.
4. The function is a linear function.
- This statement is false because a linear function has a constant rate of change, but the given function has different outputs depending on the interval.
5. The function is defined for all values of x except x=1.
- This statement is true because the function is defined for x<1 and x=1, and it is specified that when x=1, the output is 5.
Therefore, the true statements are:
- The function is discontinuous at x=1.
- The function is defined for all values of x except x=1.