Question

Math

Posted 2 months ago

```
Let $f$ be the function given by $\boldsymbol{f}(\boldsymbol{x})=\ln \frac{\boldsymbol{x}}{\boldsymbol{x}-1}$.
(a) What is the domain of $f$ ?
(b) Find the value of the derivative of $f$ at $x=-1$.
(c) Write an expression for $f^{-1}(x)$, where $f^{-1}$ denotes the inverse function of $f$.
```

Sign in to unlock the answer

Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

To find the domain of the function $f(x) = \ln \left( \frac{x}{x-1} \right)$, we need to determine where the argument of the logarithm is positive

step 2

The argument $\frac{x}{x-1}$ is positive when $x > 1$ or $x < 0$

step 3

Therefore, the domain of $f$ is $(-\infty, 0) \cup (1, \infty)$

Answer

The domain of $f$ is $(-\infty, 0) \cup (1, \infty)$.

Key Concept

Domain of a logarithmic function

Explanation

The domain of a logarithmic function is determined by the positivity of its argument.

```
Part (b)
```

step 1

To find the derivative of $f(x) = \ln \left( \frac{x}{x-1} \right)$ at $x = -1$, we use the chain rule

step 2

The derivative of $\ln \left( \frac{x}{x-1} \right)$ is $\frac{d}{dx} \left( \ln \left( \frac{x}{x-1} \right) \right) = \frac{1}{\frac{x}{x-1}} \cdot \frac{d}{dx} \left( \frac{x}{x-1} \right)$

step 3

Simplifying, we get $\frac{d}{dx} \left( \frac{x}{x-1} \right) = \frac{(x-1) \cdot 1 - x \cdot 1}{(x-1)^2} = \frac{-1}{(x-1)^2}$

step 4

Therefore, the derivative is $\frac{1}{\frac{x}{x-1}} \cdot \frac{-1}{(x-1)^2} = \frac{-1}{x(x-1)}$

step 5

Evaluating at $x = -1$, we get $\frac{-1}{-1(-1-1)} = \frac{-1}{2} = -\frac{1}{2}$

Answer

The value of the derivative of $f$ at $x = -1$ is $-\frac{1}{2}$.

Key Concept

Derivative of a logarithmic function

Explanation

The derivative of a logarithmic function can be found using the chain rule and simplifying the resulting expression.

```
Part (c)
```

step 1

To find the inverse function of $f(x) = \ln \left( \frac{x}{x-1} \right)$, we start by setting $y = \ln \left( \frac{x}{x-1} \right)$

step 2

Exponentiating both sides, we get $e^y = \frac{x}{x-1}$

step 3

Solving for $x$, we get $x = \frac{e^y}{e^y - 1}$

Answer

The inverse function of $f$ is $f^{-1}(x) = \frac{e^x}{e^x - 1}$.

Key Concept

Inverse of a logarithmic function

Explanation

The inverse of a logarithmic function can be found by exponentiating and solving for the original variable.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question