Question

Determine the equation of the inverse of (y = frac1/5e^x + 2).

a) (y = ln(frac1/5x) - 2)
b) (y = ln(frac1/5x) + 2)
c) (y = ln(5x) - 2)
d) (y = ln(5x) + 2)

Answer 1

(c)
`\(y = \ln(5x) - 2\)`

Step-by-step explanation:

To find the inverse of
`(y = (1)/(5)e^x + 2\)`, interchange
`\(x\)` and
`\(y\)` and solve for the new
`\(y\)`. Start with the original equation:

`\[ x = (1)/(5)e^y + 2 \]`

Subtract 2 from both sides:

`\[ x - 2 = (1)/(5)e^y \]`

Multiply both sides by 5 to isolate
`\(e^y\):`

`\[ 5(x - 2) = e^y \]`

Now, take the natural logarithm (ln) of both sides to solve for
`\(y\)`:

`\[ y = \ln(5x - 10) \]`

Simplify further by expressing
`\(5x - 10\)` as
`\(5(x - 2)\)`:

`\[ y = \ln(5x - 10) = \ln(5(x - 2)) \]`

Thus, the equation of the inverse function is
`\(y = \ln(5x) - 2\)`, corresponding to option (c).

Understanding the process of finding the inverse function is crucial in solving mathematical problems involving functions. The key steps involve swapping
`\(x\)` and
`\(y\)`, isolating
`\(y\),` and expressing the inverse function in terms of
`\(x\).`

Answer 2

The correct equation of the inverse of the function y = (1/5)e^x + 2 is y = ln(5x) - 2, which involves substituting and solving for y after swapping x and y in the original equation. (Option C).

Step-by-step explanation:

The inverse of the function y = \( \frac{1}{5}e^x + 2 \) can be found by first swapping y and x and then solving for y. Here's the step-by-step process:

1. Original function: y = \( \frac{1}{5}e^x + 2 \).
2. Swap x and y: x = \( \frac{1}{5}e^y + 2 \).
3. Subtract 2 from both sides: x - 2 = \( \frac{1}{5}e^y \).
4. Multiply both sides by 5: 5(x - 2) = e^y.
5. Take the natural logarithm of both sides: ln(5(x - 2)) = y. This uses the fact that ln(e^y) = y.

Thus, the correct inverse function is ln(5(x - 2)), which corresponds to answer (c) (y = ln(5x) - 2).