1. Find the domain of the composite function f o g. f(x)= 2/x-3; g(x)= 7/x
2. Solve the following exponential equation. Exact answers only. 2^1-9x = e^2x
f(g(x)) = 2/(g(x)-3) = 2/(7/x)-3) = 2x/(7-3x)
domain would be all reals except x = 7/3, but we have to recall that g(x) is not defined for x=0.
So the domain of fog is all reals except 0, 7/3
2^(1-9x) = e^2x
take log of both sides, recalling that ln(a^b) = b*lna:
(1-9x)ln2 = 2x
ln2 - 9 ln2 * x = 2x
2x + 9 ln2*x = ln2
x = ln2/(2+9 ln2)
or, if you want to be sneaky,
ln2/(ln(e^2) + ln(2^9))
= ln2/ln(512e^2)
= log512e^22
1.
fog(x) = 2 /[(7/x) - 3]
simplifying,
2x/(7 - 3x)
Domain: all real numbers except 3, 0 and 7/3
2.
2^(1-9x) = e^(2x)
get ln of both sides:
ln 2^(1-9x) = ln e^(2x)
(1-9x)*(ln 2) = 2x
(1-9x)/(2x) = 1/(ln 2)
1/(2x) - 9/2 = 1/(ln 2)
1/(2x) = 1/(ln 2) + 9/2
2x = 1/[1/(ln 2) + 4.5]
x = 1/[2(1/ln 2 + 4.5)]
x = 1/[2/(ln 2) + 9]
x = 1 / [2/(ln 2) + 9(ln 2) / (ln 2)]
x = 1 / [2 + 9(ln 2)] / ln 2
finally,
x = (ln 2)/(2 + (9 ln 2))
hope this helps~ :)
1. To find the domain of the composite function f o g, you need to consider two things:
a) The domain of the function g(x)
b) The domain of the function f(x) after substituting g(x) into it
Let's start by finding the domain of the function g(x). In this case, g(x) = 7/x. The domain of g(x) would exclude any value of x that makes the denominator equal to zero. Since the denominator is x, we know that x ≠ 0.
Next, we substitute g(x) into f(x). The composite function f o g is denoted as f(g(x)) and is equal to f(g(x)) = f(7/x). The domain of f o g will be the values of x that make both g(x) and f(g(x)) defined.
For the function f(x) = 2/(x - 3), we need to ensure that the denominator (x - 3) is not equal to zero. Solving x - 3 ≠ 0, we find x ≠ 3.
Multiplying both sides of the equation g(x) = 7/x ≠ 0 by x (assuming x ≠ 0) gives us 7 ≠ 0. This is true for all values of x.
Therefore, the domain of g(x) is all real numbers excluding 0, and the domain of f(x) with g(x) as input is all real numbers excluding 0 and 3.
Hence, the domain of the composite function f o g is all real numbers excluding 0 and 3.
2. To solve the exponential equation 2^(1-9x) = e^(2x) for x, we need to use the properties of logarithms. First, let's take the natural logarithm (ln) of both sides of the equation to eliminate the exponents:
ln(2^(1-9x)) = ln(e^(2x))
According to the logarithmic property, the exponent can come down as a coefficient:
(1-9x)ln(2) = 2xln(e)
Since ln(e) is equal to 1, we have:
(1 - 9x)ln(2) = 2x
Now, distribute ln(2) on the left side of the equation:
ln(2) - 9xln(2) = 2x
To isolate the x terms, move 2x to the left side:
-9xln(2) - 2x = -ln(2)
Once we have the x terms together, we can factor out x:
x(-9ln(2) - 2) = -ln(2)
Finally, solve for x by dividing both sides by the coefficient of x:
x = -ln(2) / (-9ln(2) - 2)
This is the exact solution to the exponential equation.