45–49. sketching graphs of functions sketch the graph of a function with the given properties. you do not need to find a formula for the function.
45. ( f(2) = 1, limlimits_{x o 2} f(x) = 3 )
46. ( f(1) = 0, f(2) = 4, f(3) = 6, limlimits_{x o 2^-} f(x) = -3, limlimits_{x o 2^+} f(x) = 5 )
47. ( g(1) = 0, g(2) = 1, g(3) = -2, limlimits_{x o 2} g(x) = 0, limlimits_{x o 3^-} g(x) = -1, limlimits_{x o 3^+} g(x) = -2 )

Question

45–49. sketching graphs of functions sketch the graph of a function with the given properties. you do not need to find a formula for the function.
45. ( f(2) = 1, limlimits_{x 	o 2} f(x) = 3 )
46. ( f(1) = 0, f(2) = 4, f(3) = 6, limlimits_{x 	o 2^-} f(x) = -3, limlimits_{x 	o 2^+} f(x) = 5 )
47. ( g(1) = 0, g(2) = 1, g(3) = -2, limlimits_{x 	o 2} g(x) = 0, limlimits_{x 	o 3^-} g(x) = -1, limlimits_{x 	o 3^+} g(x) = -2 )

Accepted Answer

To solve these problems of sketching the graphs of functions with given properties, we analyze each problem based on the definitions of function values and limits.

### Problem 45
We need to sketch a function $ f(x) $ such that $ f(2) = 1 $ and $ \lim_{x \to 2} f(x) = 3 $.

#### Step 1: Understand the function value at $ x = 2 $
The function value at $ x = 2 $ is $ f(2) = 1 $. This means the point $ (2, 1) $ is on the graph of the function.

#### Step 2: Understand the limit as $ x \to 2 $
The limit $ \lim_{x \to 2} f(x) = 3 $ means that as $ x $ approaches 2 from both the left and the right, the function values approach 3. So, we can draw a "hole" or an open circle at $ (2, 3) $ to represent the limit, and a closed dot at $ (2, 1) $ to represent the function value at $ x = 2 $. The graph can be constructed such that as $ x $ approaches 2 from either side, the function approaches 3, but at $ x = 2 $, the function has a value of 1.

### Problem 46
We need to sketch a function $ f(x) $ such that $ f(1) = 0 $, $ f(2) = 4 $, $ f(3) = 6 $, $ \lim_{x \to 2^-} f(x) = -3 $, and $ \lim_{x \to 2^+} f(x) = 5 $.

#### Step 1: Plot the function values
- At $ x = 1 $, the function value is $ f(1) = 0 $, so we plot the point $ (1, 0) $.
- At $ x = 2 $, the function value is $ f(2) = 4 $, so we plot the point $ (2, 4) $.
- At $ x = 3 $, the function value is $ f(3) = 6 $, so we plot the point $ (3, 6) $.

#### Step 2: Analyze the left - hand limit as $ x \to 2 $
The left - hand limit $ \lim_{x \to 2^-} f(x) = -3 $ means that as $ x $ approaches 2 from the left (values of $ x $ less than 2), the function values approach - 3. So, we can draw a curve approaching an open circle at $ (2, - 3) $ from the left.

#### Step 3: Analyze the right - hand limit as $ x \to 2 $
The right - hand limit $ \lim_{x \to 2^+} f(x) = 5 $ means that as $ x $ approaches 2 from the right (values of $ x $ greater than 2), the function values approach 5. So, we can draw a curve approaching an open circle at $ (2, 5) $ from the right. And we have a closed dot at $ (2, 4) $ for the function value at $ x = 2 $.

### Problem 47
We need to sketch a function $ g(x) $ such that $ g(1) = 0 $, $ g(2) = 1 $, $ g(3) = - 2 $, $ \lim_{x \to 2} g(x) = 0 $, $ \lim_{x \to 3^-} g(x) = - 1 $, and $ \lim_{x \to 3^+} g(x) = - 2 $.

#### Step 1: Plot the function values
- At $ x = 1 $, the function value is $ g(1) = 0 $, so we plot the point $ (1, 0) $.
- At $ x = 2 $, the function value is $ g(2) = 1 $, so we plot the point $ (2, 1) $.
- At $ x = 3 $, the function value is $ g(3) = - 2 $, so we plot the point $ (3, - 2) $.

#### Step 2: Analyze the limit as $ x \to 2 $
The limit $ \lim_{x \to 2} g(x) = 0 $ means that as $ x $ approaches 2 from both the left and the right, the function values approach 0. So, we draw a curve approaching an open circle at $ (2, 0) $ from both sides, and we have a closed dot at $ (2, 1) $ for the function value at $ x = 2 $.

#### Step 3: Analyze the left - hand limit as $ x \to 3 $
The left - hand limit $ \lim_{x \to 3^-} g(x) = - 1 $ means that as $ x $ approaches 3 from the left (values of $ x $ less than 3), the function values approach - 1. So, we draw a curve approaching an open circle at $ (3, - 1) $ from the left.

#### Step 4: Analyze the right - hand limit as $ x \to 3 $
The right - hand limit $ \lim_{x \to 3^+} g(x) = - 2 $ means that as $ x $ approaches 3 from the right (values of $ x $ greater than 3), the function values approach - 2. And we have a closed dot at $ (3, - 2) $ for the function value at $ x = 3 $.

For the purpose of sketching, we can summarize the key points for each function:

### For $ f(x) $ in Problem 45
- Point: $ (2,1) $ (closed dot)
- Limit behavior: Approaches $ (2,3) $ (open circle) as $ x \to 2 $ from both sides.

### For $ f(x) $ in Problem 46
- Points: $ (1,0) $, $ (2,4) $, $ (3,6) $ (closed dots)
- Limit behavior: Approaches $ (2, - 3) $ (open circle) as $ x \to 2^- $ and approaches $ (2,5) $ (open circle) as $ x \to 2^+ $

### For $ g(x) $ in Problem 47
- Points: $ (1,0) $, $ (2,1) $, $ (3, - 2) $ (closed dots)
- Limit behavior: Approaches $ (2,0) $ (open circle) as $ x \to 2 $, approaches $ (3, - 1) $ (open circle) as $ x \to 3^- $ and approaches $ (3, - 2) $ (open circle) as $ x \to 3^+ $

To actually draw the graphs:

1. For Problem 45:
   - Draw a coordinate system. Mark the point $ (2,1) $ with a closed dot.
   - Draw two curves (one from the left of $ x = 2 $ and one from the right of $ x = 2 $) that approach the open circle at $ (2,3) $.

2. For Problem 46:
   - Mark the points $ (1,0) $, $ (2,4) $, $ (3,6) $ with closed dots.
   - Draw a curve from the left of $ x = 2 $ (starting from some point left of $ x = 2 $) that approaches the open circle at $ (2, - 3) $.
   - Draw a curve from the right of $ x = 2 $ (starting from some point right of $ x = 2 $) that approaches the open circle at $ (2,5) $.
   - Connect the point $ (2,4) $ to the curve on the left and right as appropriate (since $ f(2)=4 $ is a separate point from the limit - approaching points). Also, connect $ (1,0) $ to the curve near $ x = 2^- $ and $ (3,6) $ to the curve near $ x = 2^+ $ in a way that makes the graph a function (passes the vertical line test).

3. For Problem 47:
   - Mark the points $ (1,0) $, $ (2,1) $, $ (3, - 2) $ with closed dots.
   - Draw a curve from the left of $ x = 2 $ and a curve from the right of $ x = 2 $ that both approach the open circle at $ (2,0) $.
   - Draw a curve from the left of $ x = 3 $ that approaches the open circle at $ (3, - 1) $ and a curve from the right of $ x = 3 $ that approaches the open circle at $ (3, - 2) $. Connect the points $ (1,0) $ to the curve near $ x = 2 $, $ (2,1) $ to the curves near $ x = 2 $ and $ x = 3 $, and $ (3, - 2) $ to the curve on the right of $ x = 3 $ in a way that makes the graph a function.

Since the question only asks to sketch the graphs (and we have provided the key points and the way to construct the graphs), we can conclude that we have analyzed the properties and know how to sketch the graphs for each of the given functions.

If we were to describe the final "answer" in terms of the sketching instructions:

### Answer for Problem 45
- Plot $ (2,1) $ (closed dot).
- Draw curves approaching $ (2,3) $ (open circle) as $ x \to 2 $ from left and right.

### Answer for Problem 46
- Plot $ (1,0) $, $ (2,4) $, $ (3,6) $ (closed dots).
- Draw curve approaching $ (2, - 3) $ (open circle) as $ x \to 2^- $ and curve approaching $ (2,5) $ (open circle) as $ x \to 2^+ $.

### Answer for Problem 47
- Plot $ (1,0) $, $ (2,1) $, $ (3, - 2) $ (closed dots).
- Draw curves approaching $ (2,0) $ (open circle) as $ x \to 2 $, curve approaching $ (3, - 1) $ (open circle) as $ x \to 3^- $ and curve approaching $ (3, - 2) $ (open circle) as $ x \to 3^+ $.

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QUESTION IMAGE

Question

Explanation:

Problem 45

Step 1: Understand the function value at \( x = 2 \)

Step 2: Understand the limit as \( x \to 2 \)

Problem 46

Step 1: Plot the function values

Step 2: Analyze the left - hand limit as \( x \to 2 \)

Step 3: Analyze the right - hand limit as \( x \to 2 \)

Problem 47

Step 1: Plot the function values

Step 2: Analyze the limit as \( x \to 2 \)

Step 3: Analyze the left - hand limit as \( x \to 3 \)

Step 4: Analyze the right - hand limit as \( x \to 3 \)

Answer: