The original question includes a diagram: Two increasing exponential-shaped curves through the same point on the positive y-axis; the dashed one (y=f(x)) rises more steeply than the solid one (y=ax) for x>0 and lies below it for x<0.
The graphs of two functions are shown here:
- y = ax is shown with a solid line, where a is a positive real number
- y = f(x) is shown with a dashed line
Both curves are increasing and both tend to zero as x becomes large and negative. They cross close to the y-axis, and for x > 0 the dashed curve lies above the solid one, so f grows faster than ax.
Which of the following statements (1, 2, 3, 4) could be true?
1 f(x) = bx for some b > a
2 f(x) = bx for some b < a
3 f(x) = akx for some k > 1
4 f(x) = akx for some k < 1
- A1 only
- B2 only
- C3 only
- D4 only
- E1 and 3 only
- F1 and 4 only
- G2 and 3 only
- H2 and 4 only
Show the answer and worked solution
answer · E
- A1 only
- B2 only
- C3 only
- D4 only
- E1 and 3 only
- F1 and 4 only
- G2 and 3 only
- H2 and 4 only
The solid curve is increasing, so a > 1. The dashed curve is steeper, so if f(x) = bx then b > a, making 1 possible and 2 impossible. For the other two, rewrite akx = (ak)x: this is an exponential with base ak, and since a > 1, ak > a exactly when k > 1. So 3 gives the steeper curve and is possible, while 4 gives a shallower one and is not.