Question

Choose all the key features that are true for each function (Linear, quadratic, cubic, and exponential function)

Linear f (x) = x Domain: -∞, ∞ Range: y > 0 Minimum: (0, 0) x-intercept: (0, 0) y-intercept: (0, 1)
Quadratic f (x) = x^2 Domain: -∞, ∞ Range: y > 0 Minimum: (0, 0) x-intercept: (0, 0) y-intercept: (0, 1)
Exponential f (x) = b^x, b > 1 Domain: -∞, ∞ Range: y > 0 Minimum: (0, 0) x-intercept: (0, 0) y-intercept: (0, 1)
Cubic f (x) = x^3 Domain: -∞, ∞ Range: y > 0 Minimum: (0, 0) x-intercept: (0, 0) y-intercept: (0, 1)

Answer 1

I'll mark each listed feature True or False and give the correct features where needed.

1) Linear f(x) = x
- Domain = (-∞, ∞): True
- Range = y > 0: False (Range = (-∞, ∞))
- Minimum = (0,0): False (no minimum; function is unbounded below)
- x-intercept = (0,0): True
- y-intercept = (0,1): False (y-intercept = (0,0))

2) Quadratic f(x) = x^2
- Domain = (-∞, ∞): True
- Range = y > 0: False (Range = y ≥ 0)
- Minimum = (0,0): True (global minimum value 0 at x = 0)
- x-intercept = (0,0): True (double root at 0)
- y-intercept = (0,1): False (y-intercept = (0,0))

3) Exponential f(x) = b^x, b > 1
- Domain = (-∞, ∞): True
- Range = y > 0: True
- Minimum = (0,0): False (no minimum; infimum is 0 as x → -∞ but f(x) never equals 0)
- x-intercept = (0,0): False (no x-intercepts; b^x > 0 for all x)
- y-intercept = (0,1): True (b^0 = 1)

4) Cubic f(x) = x^3
- Domain = (-∞, ∞): True
- Range = y > 0: False (Range = (-∞, ∞))
- Minimum = (0,0): False (no global minimum; odd-degree polynomial unbounded both directions)
- x-intercept = (0,0): True
- y-intercept = (0,1): False (y-intercept = (0,0))

Summary (quick):
- All four have domain (-∞, ∞).
- Only x^2 has a true minimum at (0,0).
- x, x^2, and x^3 have x- and y-intercept (0,0).
- b^x (b>1) has y-intercept (0,1), no x-intercept, and range y>0.