Question

How is the range of y = 3^x - 4 ..?

Answer 1

(-4, inf)

Answer 2

How is the range of the previously stated equation (-4, infinity)? Shouldn't it be (-3, infinity)? Since the graph of y = 3^x is moved down 4 units and y = 3^x would equal 1 when x = 0 (since all numbers to the 0th power equal 1). 1 minus 4 equals -3... not -4

Answer 3

Right, thanks :) I'm mostly wondering WHY/HOW that is...

Answer 4

well , what is the range of 3^x?

Answer 5

(1, inf) right?

Answer 6

no, it's (0,inf)

Answer 7

3^0 = 1

Answer 8

How is that?

Answer 9

if x approaches negative infinity, y approaches 0+

Answer 10

so numbers like 0.000001 is in the range, yes?

Answer 11

How could the value of y ever equal less than 1 if x^0 = 1... and the exponents can't be negative

Answer 12

I was under the impression the exponents couldn't be negative for some reason... maybe not?

Answer 13

yes, exponent CAN be negative.

for example, 3^(-2) = 1/3^2 = 1/9

Answer 14

so do you see why the range of 3^x is (0,inf)?

and since 3^x - 4 is just 4 units down, the range is (-4,inf)

Answer 15

Ohhh.. gotcha. So the domain of a^x is (-inf, inf) and the range is (0, inf)?

Answer 16

Why is the range always positive?

Answer 17

because the values of "a" are defined to be 0 < a < 1, or a > 1.

Answer 18

i.e is is positive. A positive number raised to any power is always positive

Answer 19

Oh okay, so the base cannot be negative?

Answer 20

Got it! Thanks for clearing that up

Answer 21

well, base can certainly be negative but there are many problems

for example (-3)^(1/3) is defined, but (-3)^(1/2) isn't.

Answer 22

Hmmm

Answer 23

But it's still always positive

Answer 24

the problem is if the base is negative, the function is no longer continuous . What what exponent functions to be continuous

Answer 25

no (-3)^(1/3) isn't positive

Answer 26

anyways, it has something to do with continuity. But that's higher math. For now, just now that the base is always positive, except for 1

Answer 27

Thanks a bunch :) By "higher math", what do you mean? Calculus?

Answer 28

yes, calculus or higher