FIND THE INVERSE:
f(x) = 10^(5x+3) - 10
f^-1(x) =
y = 10^(5x+3) - 10
(y+10) = 10^(5x+3)
log(y+10) = 5x+3
x = (log(y+10)-3)/5
so, f^-1(x) = (log(x+10)-3)/5
To find the inverse function, f^-1(x), we need to interchange the roles of x and y in the original equation and solve for y.
Step 1: Start with the original equation:
f(x) = 10^(5x+3) - 10
Step 2: Interchange x and y:
x = 10^(5y+3) - 10
Step 3: Solve for y:
To solve for y, we need to isolate the exponent on the right side of the equation.
First, add 10 to both sides of the equation:
10 + x = 10^(5y+3)
Next, raise both sides to the power of 1/5 to undo the exponent:
(10 + x)^(1/5) = [10^(5y+3)]^(1/5)
Simplifying the right side:
(10 + x)^(1/5) = 10^((5y+3)/5)
Since the exponent rules tell us that (a^b)^c = a^(b*c), we can rewrite the right side as:
(10 + x)^(1/5) = 10^(y + 3/5)
To get y by itself, subtract 3/5 from both sides of the equation:
(10 + x)^(1/5) - 3/5 = 10^(y + 3/5) - 3/5
Now we have y on the right side only. To isolate y, we need to undo the exponent on the right side.
Take the logarithm base 10 of both sides:
log[(10 + x)^(1/5) - 3/5] = log[10^(y + 3/5) - 3/5]
Using the logarithm rule log(a^b) = b * log(a), we can simplify the equation further:
(1/5) * log(10 + x) - (3/5) = (y + 3/5) * log(10)
Simplify the right side:
(1/5) * log(10 + x) - (3/5) = (y + 3/5) * 1
Now, simplify the equation:
(1/5) * log(10 + x) - (3/5) = y + 3/5
Finally, subtract 3/5 from both sides:
(1/5) * log(10 + x) - (3/5) - 3/5 = y
Simplify the expression on the left side:
(1/5) * log(10 + x) - 6/5 = y
Therefore, the inverse function is:
f^-1(x) = (1/5) * log(10 + x) - 6/5