Answer:
a. g(x) = 6*2^(-x)
b. y-intercept of g(x): (0,6)
c. Domain of g(x): all real numbers
d. Range of g(x): all positive real numbers
Explanation:
Main concepts:
1. Understanding the properties of exponential functions
2. Transformations of functions
3. Finding a y-intercept
4. Finding domain and range
1. Understanding the properties of exponential functions
The original function f(x)=2^x is an exponential function. Exponential functions have an equation of the form y=a*b^x where "a" is any non-zero real number, and "b" is any positive real number that is not 1.
Note, that in this case, the values of a and b are a=1 and b=2, which satisfies the rules for exponential functions.
Regardless of the values of a and b, Exponential functions always have a domain of all real numbers. The range of exponential functions is either all positive real numbers, or all negative real numbers, matching the sign of "a". Since a=1, the range of f(x) is all positive real numbers.
2. Transformations of functions
There are three categories of transformations, described by their effect on the graph of the original function:
1. A "shift" transformation (sometimes called a "translation")
2. A "stretch/compression" (where the graph is stretched or compressed toward or away from the x- or y-axis.)
3. A "reflection" (where the graph is reflected across the x- or y-axis.)
Graphically, each of these transformations can happen in one of two directions:
1. vertically (up and/or down)
2. horizontally (left and/or right)
The type of transformation is determined algebraically by the type of operation performed on the original function, and the transformation's direction is determined by its location within the function.
1. A "shift" transformation is created by adding or subtracting a number.
2. A "stretch/compression" is created by multiplying or dividing by a positive number that is greater than or less than 1.
3. A "reflection" is created by multiplying or dividing by a number that is negative.
These transformations occur vertically if the operation is outside of the core function, and occur horizontally if the operation is inside of the core function.
So, in our case, we want "reflected about the y-axis" and "stretched vertically by a factor of 6".
While the y-axis itself if the vertical axis, "reflected about the y-axis" means reflected left-right across the axis, so this would be a horizontal reflection, meaning multiplying by a negative number inside of the function.
"Stretched vertically by a factor of 6" is clearly a vertical stretch, so we'll need to multiply by a number greater than 1 on the outside of the function (specifically by 6).
Therefore, the changes yield the following new equation
f(x)=2^x
Applying the transformations to each side of the equation...
6 ( f(-x) ) =6 (2^(-x))
Since the left-side of the equation is the transformations applied to the function "f", it is the new function "g"...
g(x)=6*2^(-x)
Side note: the 6 and 2 cannot be simplified by multiplying them together as order of operations stipulates that the exponent of (-x) must be applied before multiplication is performed.
3. Finding a y-intercept
For any equation, the y-intercept has a zero-value for "x", so to find the y-intercept of this equation, substitute, zero for x, and solve for the output of the function.
g(x)=6*2^(-x)
g(0)=6*2^(-(0))
g(0)=6*2^(0)
g(0)=6*1
g(0)=6
Therefore, the function has a y-intercept of (0,6).
Graphically, this can be verified since the original function f(x) had a y-intercept of (0,1) and the function was stretched vertically by a factor of 6. Since 6 times 1 is 6, the new y-intercept is (0,6).
4. Finding domain and range
With a little algebraic manipulation, it can be determined that g(x) is itself an exponential function.
By a certain property of exponents, r^(s*t) = ((r^s)^t), g(x) can be re-written as follows:
g(x)=6*2^(-x)
g(x)=6*2^(-1*x)
g(x)=6*((2^(-1))^x)
g(x)=6*((1/2)^x)
g(x)=6*(1/2)^x
Note that this is an exponential function with a=6 and b=1/2. Therefore, the domain of g(x) is still all real numbers. Since "a" is positive, the range is still all positive real numbers.