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Sagot :
To find the range of the piecewise function [tex]\( g(x) \)[/tex], defined as
[tex]\[ g(x) = \begin{cases} x^2 - 5 & \text{if } x < 2 \\ 2x & \text{if } x \geq 2 \end{cases} \][/tex]
we will analyze each piece separately and then combine the results to obtain the overall range of the function.
### Step 1: Analyzing [tex]\( g(x) = x^2 - 5 \)[/tex] for [tex]\( x < 2 \)[/tex]
First, let's consider the part of the function for [tex]\( x < 2 \)[/tex].
- The quadratic function [tex]\( x^2 - 5 \)[/tex] is a parabola that opens upward with its vertex at the point [tex]\( x = 0 \)[/tex].
- Evaluating the function at the vertex, we find that [tex]\( g(0) = 0^2 - 5 = -5 \)[/tex].
As [tex]\( x \)[/tex] moves away from 0, either towards negative infinity or approaching [tex]\( x = 2 \)[/tex] from the left, the value of [tex]\( x^2 - 5 \)[/tex] increases.
- The function is continuous and by looking at the end values, we observe:
- As [tex]\( x \)[/tex] approaches -∞, [tex]\( g(x) = x^2 - 5 \)[/tex] approaches ∞.
- As [tex]\( x \)[/tex] approaches 2 from the left, we get [tex]\( g(2^-) = 2^2 - 5 = -1 \)[/tex].
So, for [tex]\( x < 2 \)[/tex], the minimum value of [tex]\( g(x) \)[/tex] is -5 (occuring at [tex]\( x = 0 \)[/tex]), and it can grow indefinitely large as [tex]\( x \)[/tex] moves towards -∞. Therefore, the range of this part is [tex]\((-5, \infty)\)[/tex].
### Step 2: Analyzing [tex]\( g(x) = 2x \)[/tex] for [tex]\( x \ge 2 \)[/tex]
Next, consider when [tex]\( x \geq 2 \)[/tex]:
- For [tex]\( x \ge 2 \)[/tex], the function [tex]\( g(x) = 2x \)[/tex] is a linear function with a constant positive slope of 2.
- Evaluating the function as [tex]\( x \)[/tex] increases:
- At [tex]\( x = 2 \)[/tex], [tex]\( g(2) = 2 \cdot 2 = 4 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) = 2x \)[/tex] also approaches positive infinity.
Therefore, for [tex]\( x \ge 2 \)[/tex], the range starts at 4 and extends indefinitely upwards. Thus, the range is [tex]\([4, \infty)\)[/tex].
### Step 3: Combine the ranges
To determine the overall range of the function [tex]\( g(x) \)[/tex], we combine the ranges from each piece:
- For [tex]\( x \ge 2 \)[/tex]: [4, ∞)
- For [tex]\( x < 2 \)[/tex]: [tex]\((-5, \infty)\)[/tex]
Since the pieces overlap in the interval [tex]\([4, ∞)\)[/tex], we combine them:
The overall range of the function [tex]\( g(x) \)[/tex] is [tex]\((-5, \infty)\)[/tex].
[tex]\[ g(x) = \begin{cases} x^2 - 5 & \text{if } x < 2 \\ 2x & \text{if } x \geq 2 \end{cases} \][/tex]
we will analyze each piece separately and then combine the results to obtain the overall range of the function.
### Step 1: Analyzing [tex]\( g(x) = x^2 - 5 \)[/tex] for [tex]\( x < 2 \)[/tex]
First, let's consider the part of the function for [tex]\( x < 2 \)[/tex].
- The quadratic function [tex]\( x^2 - 5 \)[/tex] is a parabola that opens upward with its vertex at the point [tex]\( x = 0 \)[/tex].
- Evaluating the function at the vertex, we find that [tex]\( g(0) = 0^2 - 5 = -5 \)[/tex].
As [tex]\( x \)[/tex] moves away from 0, either towards negative infinity or approaching [tex]\( x = 2 \)[/tex] from the left, the value of [tex]\( x^2 - 5 \)[/tex] increases.
- The function is continuous and by looking at the end values, we observe:
- As [tex]\( x \)[/tex] approaches -∞, [tex]\( g(x) = x^2 - 5 \)[/tex] approaches ∞.
- As [tex]\( x \)[/tex] approaches 2 from the left, we get [tex]\( g(2^-) = 2^2 - 5 = -1 \)[/tex].
So, for [tex]\( x < 2 \)[/tex], the minimum value of [tex]\( g(x) \)[/tex] is -5 (occuring at [tex]\( x = 0 \)[/tex]), and it can grow indefinitely large as [tex]\( x \)[/tex] moves towards -∞. Therefore, the range of this part is [tex]\((-5, \infty)\)[/tex].
### Step 2: Analyzing [tex]\( g(x) = 2x \)[/tex] for [tex]\( x \ge 2 \)[/tex]
Next, consider when [tex]\( x \geq 2 \)[/tex]:
- For [tex]\( x \ge 2 \)[/tex], the function [tex]\( g(x) = 2x \)[/tex] is a linear function with a constant positive slope of 2.
- Evaluating the function as [tex]\( x \)[/tex] increases:
- At [tex]\( x = 2 \)[/tex], [tex]\( g(2) = 2 \cdot 2 = 4 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) = 2x \)[/tex] also approaches positive infinity.
Therefore, for [tex]\( x \ge 2 \)[/tex], the range starts at 4 and extends indefinitely upwards. Thus, the range is [tex]\([4, \infty)\)[/tex].
### Step 3: Combine the ranges
To determine the overall range of the function [tex]\( g(x) \)[/tex], we combine the ranges from each piece:
- For [tex]\( x \ge 2 \)[/tex]: [4, ∞)
- For [tex]\( x < 2 \)[/tex]: [tex]\((-5, \infty)\)[/tex]
Since the pieces overlap in the interval [tex]\([4, ∞)\)[/tex], we combine them:
The overall range of the function [tex]\( g(x) \)[/tex] is [tex]\((-5, \infty)\)[/tex].
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