Explanation:
a linear function grows at a constant rate, with equal increments added or subtracted over the same interval of x, while an exponential function involves a constant multiplier that drives an increase or decrease over the same interval of x.
in our examples here the intervals of x are always 1 (1 to 2 to 3 to 4). so, we don't need to bother to norm the intervals to each other.
so, we can simply look at the differences of the f(x) data points.
in the first example the difference from 5 to 7 is 2.
the difference of 7 to 9 is 2. and the difference of 9 to 11 is 2.
so, whenever x is increased by 1, f(x) increases exactly by 2. therefore, clearly, we add the constant 2 every time.
this is a linear function.
f(x) = 2x + 3
the added constant is the factor of x, also called the slope of the line.
in the second example we see
2×2 = 4, 4×2 = 8, 8×2 = 16
so, clearly we multiply by the constant factor 2.
this is therefore an exponential function.
f(x) = 2^x
the multiplied constant is the base of the exponent.
in the third example we see that we always multiply by the constant factor ½.
½ = 0.5
0.5 × ½ = 0.25
0.25 × ½ = 0.125
0.125 × ½ = 0.0625
this is therefore an exponential function.
f(x) = (½)^x
the multiplied constant is again the base of the exponent.
in the fourth example we see that 0.03 is added constantly.
0.04 + 0.03 = 0.07
0.07 + 0.03 = 0.1
0.1 + 0.03 = 0.13
this is a linear function.
f(x) = 0.03x + 0.01
the added constant is the factor of x, also called the slope of the line.
in the fifth example we see that -4 is added constantly.
-6 - 4 = -10
-10 - 4 = -14
-14 - 4 = -18
this is a linear function.
f(x) = -4x - 2
the added constant is the factor of x, also called the slope of the line.
in the sixth example we see that there is no constant being added
1 + 15 = 16
16 + 65 = 81
but also no constant multiplication factor
1 × 16 = 16
16 × 5.0625 = 81
so, it is neither.
f(x) = x⁴
this is actuality called a polynomial (of the 4th degree).
FYI : as we can also see, a linear function is also a polynomial (of the 1st degree).