To show that the function $f: A \to \mathbb{R}$ defined by $f(x) = 5 - \frac{1}{1+x}$ for $A = \{ x \in \mathbb{R} \mid x \neq -1 \}$ is one-one, and to find its range and inverse, we proceed as follows:

Proving that $f(x)$ is One-One

A function is one-one (injective) if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. We start by assuming:

$f(x_1) = f(x_2)$

Substituting the definition of $f$:

$5 - \frac{1}{1+x_1} = 5 - \frac{1}{1+x_2}$

Subtracting 5 from both sides:

$- \frac{1}{1+x_1} = - \frac{1}{1+x_2}$

Multiplying both sides by -1:

$\frac{1}{1+x_1} = \frac{1}{1+x_2}$

Taking the reciprocals of both sides:

$1+x_1 = 1+x_2$

Subtracting 1 from both sides:

$x_1 = x_2$

Since $f(x_1) = f(x_2)$ implies $x_1 = x_2$, the function $f$ is injective.

Finding the Range of $f$

To find the range, we consider the expression $f(x) = 5 - \frac{1}{1+x}$. We examine the behavior of $f$ as $x$ approaches various values:

1. As $x \to \infty$, $\frac{1}{1+x} \to 0$ and hence $f(x) \to 5$.
2. As $x \to -1^+$, $\frac{1}{1+x} \to \infty$ and hence $f(x) \to -\infty$.
3. As $x \to -1^-$, $\frac{1}{1+x} \to -\infty$ and hence $f(x) \to \infty$.

The function $f(x)$ can take any real value because $f$ is continuous and covers all values between $-\infty$ and $5$, and $5$ itself is approached as $x \to \infty$. Therefore, the range of $f$ is:

$\text{Range of } f = (-\infty, 5)$

Finding the Inverse of $f$

To find the inverse, we start with the equation $y = 5 - \frac{1}{1+x}$ and solve for $x$:

$y = 5 - \frac{1}{1+x}$

Subtract 5 from both sides:

$y - 5 = - \frac{1}{1+x}$

Multiply both sides by -1:

$5 - y = \frac{1}{1+x}$

Take the reciprocal of both sides:

$\frac{1}{5 - y} = 1 + x$

Subtract 1 from both sides:

$x = \frac{1}{5 - y} - 1$

Therefore, the inverse function $f^{-1}$ is:

$f^{-1}(y) = \frac{1}{5 - y} - 1$

Summary

1. The function $f(x) = 5 - \frac{1}{1+x}$ is injective.
2. The range of $f$ is $(-\infty, 5)$.
3. The inverse function $f^{-1}(y)$ is $\frac{1}{5 - y} - 1$.

Would you like further details on any of these steps or have any questions?

Relative Questions

1. What does it mean for a function to be injective (one-one)?
2. How can we verify if a function is surjective (onto)?
3. What is the significance of finding the range of a function?
4. How do we derive the inverse of a function?
5. Can you explain the behavior of the function as $x \to -1$?
6. Why is the function not defined at $x = -1$?
7. What is the graphical interpretation of an inverse function?
8. How does the concept of continuity help in determining the range?

Tip

When finding the inverse of a function, always ensure that the function is one-one and onto within the specified domain and range to guarantee that the inverse function exists and is well-defined.