Show that \( f(x)=\frac{2}{x+1}+1 \)

Certainly! Let's analyze and explore the function \( f(x) = \frac{2}{x + 1} + 1 \) in detail. This analysis will cover its domain, range, asymptotes, intercepts, and graph behavior. ### 1. **Domain** The **domain** of a function consists of all real numbers \( x \) for which the function is defined. - The function \( f(x) = \frac{2}{x + 1} + 1 \) has a denominator \( x + 1 \). - Division by zero is undefined, so set the denominator not equal to zero: \[ x + 1 \neq 0 \implies x \neq -1 \] **Domain:** All real numbers except \( x = -1 \). In interval notation: \( (-\infty, -1) \cup (-1, \infty) \). --- ### 2. **Vertical Asymptote** A **vertical asymptote** occurs where the function grows without bound, typically where the denominator is zero. - From the domain restriction, \( x = -1 \) makes the denominator zero. **Vertical Asymptote:** \( x = -1 \). --- ### 3. **Horizontal Asymptote** A **horizontal asymptote** describes the behavior of the function as \( x \) approaches infinity or negative infinity. - Consider the limit of \( f(x) \) as \( x \) approaches \( \pm\infty \): \[ \lim_{x \to \pm\infty} \frac{2}{x + 1} + 1 = 0 + 1 = 1 \] **Horizontal Asymptote:** \( y = 1 \). --- ### 4. **Intercepts** #### a. **Y-Intercept** The **y-intercept** is the value of \( f(x) \) when \( x = 0 \). \[ f(0) = \frac{2}{0 + 1} + 1 = 2 + 1 = 3 \] **Y-Intercept:** \( (0, 3) \). #### b. **X-Intercept** The **x-intercept** is the value of \( x \) when \( f(x) = 0 \). Set \( f(x) = 0 \) and solve for \( x \): \[ 0 = \frac{2}{x + 1} + 1 \\ \frac{2}{x + 1} = -1 \\ 2 = - (x + 1) \quad \text{(Multiply both sides by } x + 1 \text{)} \\ 2 = -x - 1 \\ x = -3 \] **X-Intercept:** \( (-3, 0) \). --- ### 5. **Graph Behavior** - **Near the Vertical Asymptote (\( x = -1 \)):** - As \( x \) approaches \( -1 \) from the right (\( x \to -1^+ \)): \[ \frac{2}{x + 1} \to +\infty \implies f(x) \to +\infty \] - As \( x \) approaches \( -1 \) from the left (\( x \to -1^- \)): \[ \frac{2}{x + 1} \to -\infty \implies f(x) \to -\infty \] - **As \( x \) Approaches Infinity:** - The function approaches the horizontal asymptote \( y = 1 \). - **Intercepts:** - The function crosses the y-axis at \( (0, 3) \) and the x-axis at \( (-3, 0) \). ### 6. **Sketch of the Graph** While I can't provide a visual sketch here, I can describe how to plot the graph based on the information: 1. **Plot the Asymptotes:** - Draw a vertical dashed line at \( x = -1 \) (vertical asymptote). - Draw a horizontal dashed line at \( y = 1 \) (horizontal asymptote). 2. **Plot the Intercepts:** - Mark the y-intercept at \( (0, 3) \). - Mark the x-intercept at \( (-3, 0) \). 3. **Determine Behavior:** - For \( x > -1 \), the function decreases from \( +\infty \) towards the horizontal asymptote \( y = 1 \). - For \( x < -1 \), the function increases from \( -\infty \) towards the horizontal asymptote \( y = 1 \). 4. **Sketch the Curves:** - Draw two separate branches of the hyperbola, each approaching their respective asymptotes. ### 7. **Summary of Key Features** - **Domain:** \( x \in \mathbb{R} \), \( x \neq -1 \) - **Vertical Asymptote:** \( x = -1 \) - **Horizontal Asymptote:** \( y = 1 \) - **Y-Intercept:** \( (0, 3) \) - **X-Intercept:** \( (-3, 0) \) Understanding these properties allows you to graph \( f(x) \) accurately and comprehend its behavior across different intervals.