Question

Show work and explain.


`10) \: y = 10({.5})^(x)`

Growth or Decay?

y-intercept?

Domain:

Range: ​

Answer 1

Explanation:

`y=b(a)^x\\\\\text{If}\ 0<a<1,\ \text{then decay}.\\\text{If}\ a>1,\ \text{then growth.}\\\\\text{We have}\ y=10(0.5)^x\to a=0.5<1.\ \text{Therefore:}\boxed{DECAY}}\\\\y-intercept\ \text{is for x = 0. Substitute:}\\\\y=10(0.5)^0=10(1)=10\to\boxed{y-intercept=10}\\\\Domain:\text{ We can substitute any number for x,}\\\text{ because the number 0.5 can be up to any power.}\\\\\boxed{Domain=\mathbb{R}}-\text{the set of all real numbers}`

`Range:\\\text{Calculate the limits of a function:}\\\\\lim\limits_(x\to-\infty)10(0.5)^x=10\lim\limits_(x\to-\infty)\left((1)/(2)\right)^x=10\lim\limits_(x\to-\infty)2^(-x)=10(\infty)=\infty\\\\\lim\limits_(x\to\infty)10(0.5)^x=10\lim\limits_(x\to\infty)\left((1)/(2)\right)^x=10(0)=0\\\\\boxed{Range=(0,\ \infty)}`

Answer 2

Answer: Decay, y-intercept = 10, D: x is All Real Numbers, R: y > 0

Explanation:

y = 10(0.5)ˣ

↓

represents the y-intercept (when x = 0, y = 10(0.5)⁰ = 10)

y = 10(0.5)ˣ

↓

If >1, represents growth. If <1, represents decay.

Domain: There are no restrictions on x so x is All Real Numbers (-∞, ∞)

Range: No matter what value x is, y has to be greater than 0 (0, ∞)

We can also show this algebraically:

y = 10(0.5)ˣ

log y = log 10(0.5)ˣ

↓

y > 0 because you cannot take the log of 0 or a negative value

See graph