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Sagot :
Sure, let's address the given questions step-by-step.
a) Domain and Range of the function [tex]\( f(x) = |x-3| + 1 \)[/tex]
Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For the function [tex]\( f(x) = |x-3| + 1 \)[/tex], the absolute value function [tex]\( |x-3| \)[/tex] is defined for all real numbers. Therefore, there are no restrictions on the values of [tex]\( x \)[/tex]. The function can accept any real number as input.
Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
Range:
The range of a function is the set of all possible output values (y-values).
To determine the range, let's analyze the behavior of [tex]\( f(x) = |x-3| + 1 \)[/tex]:
- The absolute value [tex]\( |x-3| \)[/tex] is always greater than or equal to 0.
- The smallest value [tex]\( |x-3| \)[/tex] can take is 0, which occurs when [tex]\( x = 3 \)[/tex].
When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = |3-3| + 1 = 0 + 1 = 1 \][/tex]
As [tex]\( x \)[/tex] moves away from 3 in either direction (positive or negative), [tex]\( |x-3| \)[/tex] increases, and hence [tex]\( f(x) \)[/tex] also increases.
Therefore, the range of [tex]\( f(x) \)[/tex] starts at 1 and increases without bound.
Thus, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Range} = [1, \infty) \][/tex]
b) Sketch of the graph of [tex]\( y = f(x) \)[/tex]
To sketch the graph of [tex]\( f(x) = |x-3| + 1 \)[/tex], we can follow these steps:
1. Identify key points:
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 1 \)[/tex].
- As [tex]\( x \)[/tex] moves away from 3, [tex]\( f(x) \)[/tex] increases linearly.
2. Behavior on either side of [tex]\( x = 3 \)[/tex]:
- For [tex]\( x < 3 \)[/tex], [tex]\( f(x) = 3 - x + 1 = 4 - x \)[/tex]. This is a linear function with a negative slope.
- For [tex]\( x > 3 \)[/tex], [tex]\( f(x) = x - 3 + 1 = x - 2 \)[/tex]. This is a linear function with a positive slope.
3. Plotting points and drawing the graph:
- Plot the point [tex]\( (3, 1) \)[/tex] where the minimum value occurs.
- For [tex]\( x < 3 \)[/tex], draw a line with a negative slope.
- For [tex]\( x > 3 \)[/tex], draw a line with a positive slope.
The graph can be illustrated as follows:
```
y
|
5 | /
4 | /
3 | /
2 | /
1 |----------o---------
| /
-5 | /
-10 0 10
x
```
- The point [tex]\( (3, 1) \)[/tex] is the vertex of the V-shaped graph.
- To the left of [tex]\( x = 3 \)[/tex], the line decreases with a slope of -1.
- To the right of [tex]\( x = 3 \)[/tex], the line increases with a slope of 1.
Thus, we've identified both the domain and range of the function and sketched a graph of [tex]\( y = f(x) \)[/tex].
a) Domain and Range of the function [tex]\( f(x) = |x-3| + 1 \)[/tex]
Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For the function [tex]\( f(x) = |x-3| + 1 \)[/tex], the absolute value function [tex]\( |x-3| \)[/tex] is defined for all real numbers. Therefore, there are no restrictions on the values of [tex]\( x \)[/tex]. The function can accept any real number as input.
Thus, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
Range:
The range of a function is the set of all possible output values (y-values).
To determine the range, let's analyze the behavior of [tex]\( f(x) = |x-3| + 1 \)[/tex]:
- The absolute value [tex]\( |x-3| \)[/tex] is always greater than or equal to 0.
- The smallest value [tex]\( |x-3| \)[/tex] can take is 0, which occurs when [tex]\( x = 3 \)[/tex].
When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = |3-3| + 1 = 0 + 1 = 1 \][/tex]
As [tex]\( x \)[/tex] moves away from 3 in either direction (positive or negative), [tex]\( |x-3| \)[/tex] increases, and hence [tex]\( f(x) \)[/tex] also increases.
Therefore, the range of [tex]\( f(x) \)[/tex] starts at 1 and increases without bound.
Thus, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Range} = [1, \infty) \][/tex]
b) Sketch of the graph of [tex]\( y = f(x) \)[/tex]
To sketch the graph of [tex]\( f(x) = |x-3| + 1 \)[/tex], we can follow these steps:
1. Identify key points:
- At [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 1 \)[/tex].
- As [tex]\( x \)[/tex] moves away from 3, [tex]\( f(x) \)[/tex] increases linearly.
2. Behavior on either side of [tex]\( x = 3 \)[/tex]:
- For [tex]\( x < 3 \)[/tex], [tex]\( f(x) = 3 - x + 1 = 4 - x \)[/tex]. This is a linear function with a negative slope.
- For [tex]\( x > 3 \)[/tex], [tex]\( f(x) = x - 3 + 1 = x - 2 \)[/tex]. This is a linear function with a positive slope.
3. Plotting points and drawing the graph:
- Plot the point [tex]\( (3, 1) \)[/tex] where the minimum value occurs.
- For [tex]\( x < 3 \)[/tex], draw a line with a negative slope.
- For [tex]\( x > 3 \)[/tex], draw a line with a positive slope.
The graph can be illustrated as follows:
```
y
|
5 | /
4 | /
3 | /
2 | /
1 |----------o---------
| /
-5 | /
-10 0 10
x
```
- The point [tex]\( (3, 1) \)[/tex] is the vertex of the V-shaped graph.
- To the left of [tex]\( x = 3 \)[/tex], the line decreases with a slope of -1.
- To the right of [tex]\( x = 3 \)[/tex], the line increases with a slope of 1.
Thus, we've identified both the domain and range of the function and sketched a graph of [tex]\( y = f(x) \)[/tex].
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