problem 12: consider the graph of a function below.
(a) find f(0) and f(6).
(b) find f(2) and f(-2).
(c) is f(3) positive or negative?
(d) is f(-1) positive or negative?
(e) for what values of x is f(x) = 0?
(f) for what values of x is f(x)

Question

problem 12: consider the graph of a function below.
(a) find f(0) and f(6).
(b) find f(2) and f(-2).
(c) is f(3) positive or negative?
(d) is f(-1) positive or negative?
(e) for what values of x is f(x) = 0?
(f) for what values of x is f(x) < 0?
(g) what is the domain of f?
(h) what is the range of f?
(i) what are the x-intercepts?
(j) what is the y-intercept?
(k) how often does the line y = -1 intersect the graph?
(l) how often does the line x = 1 intersect the graph?
(m) for what value of x does f(x) = 3?
(n) for what value of x does f(x) = -2?

Accepted Answer

### Part (a)
#### Step 1: Find $ f(0) $
To find $ f(0) $, we look at the point where $ x = 0 $ on the graph. The graph passes through $ (0, 0) $, so $ f(0) = 0 $.

#### Step 2: Find $ f(6) $
To find $ f(6) $, we look at the point where $ x = 6 $ on the graph. The graph passes through $ (6, 0) $, so $ f(6) = 0 $.

### Part (b)
#### Step 1: Find $ f(2) $
To find $ f(2) $, we look at the point where $ x = 2 $ on the graph. The graph passes through $ (2, -2) $, so $ f(2) = -2 $.

#### Step 2: Find $ f(-2) $
To find $ f(-2) $, we look at the point where $ x = -2 $ on the graph. The graph passes through $ (-2, 1) $, so $ f(-2) = 1 $.

### Part (c)
#### Step 1: Analyze $ f(3) $
We look at the graph for $ x = 3 $. The interval $ 2 < x < 4 $ (where $ x = 3 $ lies) has the graph below the x - axis (since at $ x = 2 $, $ y=-2 $ and at $ x = 4 $, $ y = 0 $, and the graph is a curve from $ (2,-2) $ to $ (4,0) $). So $ f(3) $ is negative.

### Part (d)
#### Step 1: Analyze $ f(-1) $
We look at the graph for $ x=-1 $. The interval $ -2 < x < 0 $ (where $ x = -1 $ lies) has the graph above the x - axis (since at $ x=-2 $, $ y = 1 $ and at $ x = 0 $, $ y=0 $, and the graph is a curve from $ (-2,1) $ to $ (0,0) $). So $ f(-1) $ is positive.

### Part (e)
#### Step 1: Find $ x $ where $ f(x)=0 $
We look for the x - intercepts (points where $ y = 0 $) on the graph. The points are $ (0,0) $, $ (4,0) $, and $ (6,0) $. So $ x = 0 $, $ x = 4 $, and $ x = 6 $.

### Part (f)
#### Step 1: Find $ x $ where $ f(x)<0 $
We look for the intervals where the graph is below the x - axis ( $ y<0 $ ). The graph is below the x - axis between $ 0 < x < 4 $ (since at $ x = 0 $, $ y = 0 $; at $ x = 2 $, $ y=-2 $; at $ x = 4 $, $ y = 0 $). So $ 0 < x < 4 $.

### Part (g)
#### Step 1: Find the domain of $ f $
The domain is the set of all x - values for which the function is defined. The leftmost point is $ (-4,2) $ and the rightmost point is $ (6,0) $. So the domain is $ -4\leq x\leq6 $ or $ [-4,6] $.

### Part (h)
#### Step 1: Find the range of $ f $
The range is the set of all y - values the function takes. The lowest y - value is $ -2 $ (at $ x = 2 $) and the highest y - value is $ 3 $ (at $ x = 5 $). So the range is $ -2\leq y\leq3 $ or $ [-2,3] $.

### Part (i)
#### Step 1: Find the x - intercepts
The x - intercepts are the points where $ y = 0 $. From the graph, the points are $ (0,0) $, $ (4,0) $, and $ (6,0) $.

### Part (j)
#### Step 1: Find the y - intercept
The y - intercept is the point where $ x = 0 $. From the graph, the point is $ (0,0) $, so the y - intercept is $ 0 $.

### Part (k)
#### Step 1: Find intersections with $ y=-1 $
We draw the line $ y=-1 $ and see how many times it intersects the graph. The graph between $ 0 < x < 4 $ (the curve from $ (0,0) $ to $ (4,0) $ passing through $ (2,-2) $) will intersect $ y=-1 $ twice (once between $ 0 < x < 2 $ and once between $ 2 < x < 4 $). Also, the graph between $ 4 < x < 6 $ is a triangle - like shape from $ (4,0) $ to $ (5,3) $ to $ (6,0) $, and it does not intersect $ y=-1 $ (since the minimum y - value there is $ 0 $). The graph between $ -4 < x < 0 $ is above $ y=-1 $ (since the minimum y - value there is $ 0 $ at $ x = 0 $ and $ y = 1 $ at $ x=-2 $). So the line $ y=-1 $ intersects the graph twice.

### Part (l)
#### Step 1: Find intersections with $ x = 1 $
We draw the line $ x = 1 $ (a vertical line) and see how many times it intersects the graph. The graph between $ 0 < x < 4 $ (the curve from $ (0,0) $ to $ (4,0) $ passing through $ (2,-2) $) will be intersected by $ x = 1 $ once (since $ x = 1 $ is between $ 0 $ and $ 4 $). The other parts of the graph (left of $ x = 0 $ and right of $ x = 4 $) do not intersect $ x = 1 $. So the line $ x = 1 $ intersects the graph once.

### Part (m)
#### Step 1: Find $ x $ where $ f(x)=3 $
We look for the point where $ y = 3 $ on the graph. The point is $ (5,3) $, so $ x = 5 $.

### Part (n)
#### Step 1: Find $ x $ where $ f(x)=-2 $
We look for the point where $ y=-2 $ on the graph. The point is $ (2,-2) $, so $ x = 2 $.

### Answers:
(a) $ f(0)=\boldsymbol{0} $, $ f(6)=\boldsymbol{0} $

(b) $ f(2)=\boldsymbol{-2} $, $ f(-2)=\boldsymbol{1} $

(c) $ f(3) $ is $\boldsymbol{\text{negative}}$

(d) $ f(-1) $ is $\boldsymbol{\text{positive}}$

(e) $ x=\boldsymbol{0,4,6} $

(f) $ \boldsymbol{0 < x < 4} $

(g) Domain: $\boldsymbol{[-4,6]}$

(h) Range: $\boldsymbol{[-2,3]}$

(i) x - intercepts: $\boldsymbol{(0,0),(4,0),(6,0)}$

(j) y - intercept: $\boldsymbol{0}$

(k) Intersects $\boldsymbol{2}$ times

(l) Intersects $\boldsymbol{1}$ time

(m) $ x=\boldsymbol{5} $

(n) $ x=\boldsymbol{2} $

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Find \( f(0) \)

Step 2: Find \( f(6) \)

Part (b)

Step 1: Find \( f(2) \)

Step 2: Find \( f(-2) \)

Part (c)

Step 1: Analyze \( f(3) \)

Part (d)

Step 1: Analyze \( f(-1) \)

Part (e)

Step 1: Find \( x \) where \( f(x)=0 \)

Part (f)

Step 1: Find \( x \) where \( f(x)<0 \)

Part (g)

Step 1: Find the domain of \( f \)

Part (h)

Step 1: Find the range of \( f \)

Part (i)

Step 1: Find the x - intercepts

Part (j)

Step 1: Find the y - intercept

Part (k)

Step 1: Find intersections with \( y=-1 \)

Part (l)

Step 1: Find intersections with \( x = 1 \)

Answer: