The solutions to the questions are:
- The equation of the function is g(x) = -7/8(2^-5x) + 7
- The domain is -∝ < x < ∝ while the range is y < 7
- The horizontal asymptote is y = 7 and the function is an example of exponential decay
- The y-intercept is 49/8 while the x-intercept is -0.6
<h3>State the parent function in the form and express the function in the form</h3>
The function is an exponential function.
So, the parent function has the form
y = ab^x
Using the function in (a), the parent function is: y = -7/8 * 2^x and the form of the function is g(x) = -7/8(2^-5x) + 7
<h3>The transformation</h3>
First, the function is horizontally stretched by -5
This gives
g(x) = -7/8(2^-5x)
First, the function is shifted up by 7 units
This gives
g(x) = -7/8(2^-5x) + 7
<h3>Express the function in the form y = ab^x + c</h3>
The equation of the function is given as:
g(x) = -28/[2^(5x + 5)] + 7
Rewrite the equation as follows:
g(x) = -28/[2^(5x) * 2^5] + 7
Evaluate the exponent
g(x) = -28/[2^(5x) * 32] + 7
Divide
g(x) = -7/[2^(5x) * 8] + 7
Rewrite as:
g(x) = -7/[8 * 2^(5x)] + 7
Further, rewrite as:
g(x) = -7/8 * 2^(-5x) + 7
Rewrite properly as:
<h3>Determine any asymptotes and state whether the function is an example of exponential growth or decay</h3>
We have:
g(x) = -7/8 * 2^(-5x) + 7
Set the radical to 0
g(x) = 0 + 7
Evaluate
g(x) = 7
This represents the horizontal asymptote (it has no vertical asymptote)
Hence, the horizontal asymptote is y = 7 and the function is an example of exponential decay
<h3>Determine the domain and range of the function.</h3>
The function can take any input
So, the domain is -∝ < x < ∝
We have the horizontal asymptote to be
y = 7
The function cannot equal or exceed this value.
So, the range is y < 7
<h3>Calculate the x-intercept and y-intercept, then sketch the function.</h3>
Set x = 0
g(0) = -7/8 * 2^(-5 * 0) + 7
This gives
g(0) = -7/8 * 2^(0) + 7
Evaluate the exponent
g(0) = -7/8 + 7
Evaluate the sum
g(0) = 49/8
So, the y-intercept is 49/8
Set g(x) = 0
0 = -7/8 * 2^(-5x) + 7
This gives
-7 = -7/8 * 2^(-5x)
Divide by -7
1 = 1/8 * 2^(-5x)
Multiply by 8
8 = 2^(-5x)
Solve for x
x = -0.6
So, the x-intercept is -0.6
See attachment for the sketch
Read more about exponential functions at:
brainly.com/question/2456547
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