solve the equation
1. In x + In (x+12) =2
x= ? or either there is no solution
2. solve the exponential equation
9^x=6
What is the exact answer?
3. Use transformations to indentify the graph of the function. Then determine its domain, range, and horizontal asymptote.
f(x)=2^-x+1
x=? or either there is no solution
1. To solve the equation In x + In (x+12) = 2, we can use properties of logarithms to simplify the equation.
First, we can combine the natural logarithms using the property ln(a) + ln(b) = ln(a*b):
ln(x) + ln(x+12) = ln(x(x+12)) = ln(x^2 + 12x) = 2.
Next, we can convert the equation to exponential form, where the base is e:
e^2 = x^2 + 12x.
We now have a quadratic equation. Rearranging the terms:
x^2 + 12x - e^2 = 0.
This quadratic equation can be solved using factoring, completing the square, or using the quadratic formula. Since we have a general equation, we cannot solve for x accurately without knowing the value of e.
Therefore, the answer to the equation x = ? or either there is no solution, as we do not have enough information.
2. To solve the exponential equation 9^x = 6, we can use logarithms to isolate x.
Taking the natural logarithm (ln) of both sides of the equation:
ln(9^x) = ln(6).
Using the property of logarithms, we can bring the exponent x down:
x * ln(9) = ln(6).
Now, we can isolate x by dividing both sides by ln(9):
x = ln(6) / ln(9).
Using a calculator to evaluate the natural logarithms, we find that x is approximately 0.4918.
So, the exact answer for the exponential equation 9^x = 6 is x ≈ 0.4918.
3. To identify the graph of the function f(x) = 2^(-x+1), we can analyze the transformations from the parent function 2^x.
First, we notice a horizontal shift of 1 unit to the right, as indicated by the "+1" in the exponent. This means that the graph of f(x) is shifted to the right compared to the parent function.
Next, we see a reflection across the x-axis, indicated by the negative sign in front of the x-term. This means that the graph of f(x) is the mirror image of the parent function about the x-axis.
Finally, we note that the base of the exponential function is 2, which means the graph will have a vertical asymptote at y = 0.
The domain of the function is all real numbers, since the exponential function is defined for all x.
The range of the function is all positive real numbers except for 0, since the graph is a reflection of the parent function.
Therefore, the domain of the function f(x) = 2^(-x+1) is all real numbers, the range is (0, ∞), and the horizontal asymptote is y = 0.