Question

What is the range of f?

mc002-1.jpg

mc002-2.jpg

mc002-3.jpg

all real numbers

Answer 1

I can’t see the images you attached — could you either re-upload them or type the function/describe the graph? I’ll need that to give the exact range.

Meanwhile, here are quick methods to find a function’s range depending on how it’s given:

- If you have a graph: read off the y-values attained by the curve. The range is the interval (or union of intervals) of y-values the graph hits. Pay attention to endpoints and arrows (→ means extends to ±∞). Holes or horizontal asymptotes can remove single y-values.

- If you have an algebraic formula:
- Linear f(x)=mx+b: range is all real numbers.
- Quadratic f(x)=ax^2+bx+c: if a>0, range is [vertex y, ∞); if a<0, range is (−∞, vertex y]. Vertex y = c − b^2/(4a) (or use completing the square).
- Square root f(x)=√(g(x)): range is [minimum of √-expression, ∞) (nonnegative outputs).
- Exponential f(x)=A e^{kx}+C: range is (C, ∞) if A>0, (−∞, C) if A<0.
- Logarithm f(x)=ln(g(x))+C: range is (−∞, ∞).
- Rational function f(x) = (ax+b)/(cx+d): typically all real y except the horizontal asymptote y = a/c (if c≠0), but check for values excluded by solving for x.
- To find range algebraically: try to solve y = f(x) for x and see which y produce a real x. Also use derivatives to find extrema.

If you re-upload the images or paste the function I’ll compute the range exactly.