find the domain
f(x)=x/x-8,g(x)=-7/x
solve the following exponential equation equation exact answers only
3e x-1 = 2+e^x-1
To find the domain of a function, you need to determine the values of x for which the function is defined. In other words, you need to identify any restrictions or limitations on the values that x can take.
For the function f(x) = x/(x - 8), the only restriction is that x cannot equal 8. If x were 8, the denominator would be zero, and division by zero is undefined. Therefore, the domain of f(x) is all real numbers except 8.
For the function g(x) = -7/x, the only restriction is that x cannot equal 0. Division by zero is undefined, so x cannot be zero. Therefore, the domain of g(x) is all real numbers except 0.
Now let's move on to the exponential equation:
3e^(x - 1) = 2 + e^(x - 1)
To solve this equation for x, we can simplify it step by step:
Step 1: Divide both sides of the equation by e^(x - 1):
3 = (2 + e^(x - 1))/e^(x - 1)
Step 2: Simplify the right side of the equation by using the properties of exponents:
3 = 2/e^(x - 1) + (e^(x - 1))/e^(x - 1)
= 2/e^(x - 1) + 1
Step 3: Subtract 1 from both sides:
3 - 1 = 2/e^(x - 1)
Step 4: Simplify the left side:
2 = 2/e^(x - 1)
Step 5: Cross-multiply:
2e^(x - 1) = 2
Step 6: Divide both sides by 2:
e^(x - 1) = 1
Step 7: Take the natural logarithm (ln) of both sides to remove the exponential:
ln(e^(x - 1)) = ln(1)
(x - 1)ln(e) = 0
Since ln(e) = 1, the equation becomes:
(x - 1) = 0
Step 8: Solve for x:
x - 1 = 0
x = 1
Therefore, the exact solution to the exponential equation 3e^(x - 1) = 2 + e^(x - 1) is x = 1.