Question

Define the range of the following function.
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all real numbers

Answer 1

I can’t see the function from your message. Did you mean to say the answer is “all real numbers” and want that checked?

Please either paste the function here or paste the image description. If you want to check whether a function has range all real numbers, these are the usual steps:

- If it’s a polynomial:
- odd degree with real coefficients → range = all real numbers.
- even degree → check leading coefficient and find global minimum/maximum (via derivative) to get a half-line range like [m, ∞) or (−∞, M].
- If it’s a rational function f(x)=p(x)/q(x):
- examine horizontal/oblique asymptotes (limits as x→±∞) and any y-values excluded by vertical asymptotes or holes.
- If it’s composed of exponentials, logs, trig, roots, etc., use known ranges (e.g., e^x > 0, ln x real domain x>0, sin/cos in [−1,1]) and algebraic manipulation to see attainable y.
- In all cases you can:
1. Solve y = f(x) for x in terms of y and determine which y produce real x.
2. Take limits as x→±∞ to see end behavior.
3. Find critical points (f′(x)=0) to locate extrema.

Send the function and I’ll compute the exact range and verify whether it is indeed all real numbers.