Mr. IntegralFunctions That Are Not Onto
Identify codomain values that a function does not reach.
The bigger question: What does a function tell us—and what can it hide?
On this page
Idea
Some texts call a function “into” when its range is a proper subset of its codomain. Because terminology varies, “not onto” is the clearer statement. Such a function leaves at least one declared target unreachable.
Visual guide
Range is smaller than the codomain
Method
To disprove onto behavior, exhibit a single element of the codomain with no preimage. You must know the codomain before making this judgment.
Worked example
For with , the target cannot be reached by a real input. Thus is not onto. If instead the declared codomain is , every target has preimage , so the same formula and domain become onto that codomain.
Try it yourself.
Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.
Hint 1 · Find a starting point
Onto depends on the declared codomain.
Hint 2 · Take the next step
The codomain is ℝ but the actual outputs are positive.
Show the reasoning
Answer: It never produces a nonpositive output.
For example, no real x satisfies eˣ=0, so some codomain values are missed.
Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.
Common mistake
A function can be one-to-one without being onto, or onto without being one-to-one. These are independent properties.
Check your understanding
Is onto?
Show answer
No. Zero and all negative targets are unreachable; its range is .
Pause before the next idea.
Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.