 Sketch the graph of a function f with all of the following properties.
• f (x) < 2 for all x
• f is increasing on its domain (−∞, ∞)
• f is concave up on (−∞, −1)

To sketch the graph of the function \( f \) with the given properties, let's go through each condition step-by-step:

1. **\( f(x) < 2 \) for all \( x \):**
   - This means the function never reaches or exceeds the value \( y = 2 \). It can approach this value but will always remain below it.

2. **\( f \) is increasing on its domain \( (-\infty, \infty) \):**
   - An increasing function means that as \( x \) increases, \( f(x) \) also increases. So, the slope of \( f \) is always positive.

3. **\( f \) is concave up on \( (-\infty, -1) \):**
   - Concave up means the second derivative \( f''(x) > 0 \) on this interval. The graph will look like a "U" shape on \( (-\infty, -1) \), opening upwards.

### Sketching the Graph:
- Start by sketching a curve that remains entirely below the horizontal line \( y = 2 \). This is an upper boundary that the function approaches but does not touch or cross.
- Since \( f \) is increasing over the entire domain \( (-\infty, \infty) \), the graph should rise from left to right.
- From \( (-\infty, -1) \), the function is concave up, so it should look like the left side of a parabola opening upwards.
- After \( x = -1 \), while not specified, we can assume a smooth transition where the function remains increasing but no longer has to be concave up or down. A simple choice is that the function becomes a horizontal asymptote as it approaches \( y = 2 \).

### Basic Sketch Outline:
1. Draw a horizontal asymptote at \( y = 2 \).
2. Sketch a curve that starts at the left side of the graph, increasing and concave up as it moves towards \( x = -1 \).
3. As the curve passes \( x = -1 \), continue to draw it so it keeps increasing and approaches the asymptote \( y = 2 \) without crossing it.

Would you like a detailed step-by-step sketch of this graph or more explanation on how to handle different parts of the function?

### Follow-up Questions:
1. What happens to the graph if the function \( f \) is concave down on \( (-1, \infty) \)?
2. How would the graph look if \( f \) had a point of inflection at \( x = -1 \)?
3. What other functions satisfy all three given properties?
4. Can you derive a specific function formula that meets these properties?
5. How would the behavior change if the condition \( f(x) < 2 \) is removed?

### Tip:
When sketching graphs with asymptotic behavior, it can be helpful to first draw the asymptote as a guide to ensure the function approaches it correctly.

Sketch the graph of a function f with all of the following properties.
• f(x) < 2 for all x
• f is increasing on its domain (−∞, ∞)
• f is concave up on (−∞, −1)

Learn how to sketch the graph of a function that is always increasing, concave up on (-∞, -1), and has a horizontal asymptote at y = 2. This detailed solution covers key concepts such as concavity, increasing functions, and asymptotic behavior, suitable for high school students in Grades 11-12.

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