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Answer:
The domain of h(x) is {x : x ∈ R}
The range of the function is {y : y > -9}
The horizontal asymptote is at y = -9
Step-by-step explanation:
* Lets read the problem and solve it
- The exponential function is f(x) = a(b)^x, where a and b are constant
and b is the base , x is the exponent , a is the initial value of f(x)
- The domain of the function is all the values of x which make the
function defined
- The range of the function is the set of values of y that corresponding
with the domain x
- Asymptote on the graph a line which is approached by a curve but
never reached
- A function of the form f(x) = a(b^x) + c always has a horizontal
asymptote at y = c
* Lets solve the problem
∵ h(x) = (0.5)^x - 9
∵ All the values of x make h(x) defined
∵ The domain of the function is the values of x
∴ The domain of h(x) is {x : x ∈ R} ⇒ R is the set of real number
∵ The range of the function is the set of values of y which
corresponding to x
∵ (0.5)^x must be positive because there is no values of x make it
negative value
∴ y must be greater than -9
∴ The range of the function is {y : y > -9}
∵ A function of the form f(x) = a (bx) + c always has a horizontal
asymptote at y = c
∵ h(x) = (0.5)^x - 9
∴ c = -9
∴ The horizontal asymptote is at y = -9
* Look to the attached file for more understanding
Given that the jeweler has alloys that are 25% gold, 50% gold, and 82% gold.
As he wants to make 14 grams of an alloy by adding two different alloys that is precisely 75% gold, so one alloy must have a percentage of gold more than 75%.
One alloy is 82% gold and, the second can be chosen between 25% gold, 50% gold, so there are two cases.
Case 1: 82% gold + 50% gold
Let x grams of 82% gold and y grams of 50% gold added to make x+y=14 grams of 75% gold, so
75% of 14 = 82% of x + 50% of y
[as x+y=14]
grams
and grams.
Hence, 10.9375 grams of 82% gold and 3.0625 grams of 50% gold added to make 14 grams of 75% gold.
Case 2: 82% gold + 25% gold
Let x grams of 82% gold and y grams of 25% gold added to make x+y=14 grams of 75% gold, so
75% of 14 = 82% of x + 25% of y
grams
and grams.
Hence, 12.28 grams of 82% gold and 1.72 grams of 50% gold added to make 14 grams of 75% gold.