Functions and Relations Question
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Grade 11 Functions and Relations CCSS: HSF-LE.A.3
The following table shows some of the values of functions f(x), which is linear, and g(x), which is exponential. What can be concluded from these values (and general properties of these functions), and why?
| x | f(x) | g(x) |
| 2 | 23 | 4 |
| 3 | 33 | 8 |
| 4 | 43 | 16 |
| 5 | 53 | 32 |
| 6 | 63 | 64 |
| 7 | 73 | 128 |
- g(x)>f(x) for all values of x. This is because f(x) is an exponential function, which must always be greater than a linear function.
- g(x)>f(x) for x≥6. Since the values of g(x) continue to increase by ever greater amounts each unit interval, they will continue to be greater than the values of f(x) which are only increasing by the same amount each unit interval.
- g(x)>f(x) for 6≤x≤45. If a linear function intersects an exponential function, it must do so twice. Therefore, the two functions will intersect again at approximately x≈25.5≈45.3, and thereafter f(x)>g(x).
- g(x)>f(x) for x≥6 and g(x)<f(x) for x≤5. Since x≈5.5 is the only point of intersection, and given the values in the table, g(x) must be greater than f(x) for values of x greater than or equal to 6 and it must be less than f(x) for values of x less than or equal to 5.