I need help finding the inverse of f(x)=7^3x-5 with work shown.
your f(x)=7^3x-5 is ambiguous
did you mean it the way your typed it or
f(x)=7^(3x-5)
The way you typed it:
let y = 7^3x - 5
y + 5 = 7^(3x) , another ambiguity, the way you typed it, it would be
(7^3)(x)
I will go with my last interpretation.
y + 5 = 7^(3x)
step 1 is to interchange the x and y variables , so we get
x + 5 = 7^(3y)
step 2 is to solve this new equation for y, this requires logs
log(x+5) = log ( 7^(3x) )
log(x+5) = 3x(log7)
x = log(x+5)/(3log7) , x > 5
if you meant y = 7^(3x-5)
then we get x = 7^(3y-5), and now take logs of that etc.
jgip87opgpy8iu
To find the inverse of a function, f(x), we need to interchange the x and y variables and solve for y.
Given f(x) = 7^(3x-5), let's start by interchanging the x and y variables:
x = 7^(3y - 5)
Next, we solve for y. To do this, we need to isolate the exponential term:
7^(3y - 5) = x
Taking the logarithm of both sides of the equation will help us isolate the exponent:
log base 7 (7^(3y - 5)) = log base 7 (x)
Using the logarithm property, we can bring the exponent to the front:
(3y - 5) log base 7 (7) = log base 7 (x)
Since log base 7 (7) equals 1, the equation simplifies to:
3y - 5 = log base 7(x)
Now, isolate the y variable by adding 5 to both sides of the equation:
3y - 5 + 5 = log base 7(x) + 5
3y = log base 7(x) + 5
Finally, divide both sides of the equation by 3 to solve for y:
(3y)/3 = (log base 7(x) + 5)/3
y = (1/3) * (log base 7(x) + 5)
So, the inverse function of f(x) = 7^(3x-5) is:
f^(-1)(x) = (1/3) * (log base 7(x) + 5)
To find the inverse of a function, a common approach is to switch the x and y variables and then solve for y.
Step 1: Replace f(x) with y:
y = 7^(3x) - 5
Step 2: Swap the x and y variables:
x = 7^(3y) - 5
Step 3: Solve the equation for y.
Let's go through the steps to find the inverse:
Step 1: Add 5 to both sides of the equation to isolate the exponential term:
x + 5 = 7^(3y)
Step 2: Take the logarithm (log base 7) of both sides to eliminate the exponential term:
log₇(x + 5) = 3y
Step 3: Divide both sides by 3 to solve for y:
y = (1/3) * log₇(x + 5)
Therefore, the inverse of f(x) = 7^(3x) - 5 is:
f^(-1) (x) = (1/3) * log₇ (x + 5)
Please note that in order for the inverse function to be well-defined, certain restrictions may apply in terms of the domain and range of the original function.