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Answer:
1. Start with the basic function: The function ( y = x² ) represents a parabola that opens upwards with its vertex at (0, 0).
2. Shift the function: The given function is ( y = x² + 1 ), which means the entire parabola is shifted 1 unit up. Therefore, its vertex is now at (0, 1).
3. Determine the minimum value: The minimum value of ( y = x² + 1 ) occurs at the vertex of the parabola. Since ( x² ) is always non-negative
[tex](i.e., ( x^2 \geq 0 ))[/tex]
the smallest value of ( y ) is:
[tex]y_{\text{min}} = 0 + 1 = 1[/tex]
4. Determine the range: As ( x ) increases or decreases without bound, ( x² ) becomes very large, and so does ( y ). Therefore, ( y ) can take any value greater than or equal to 1.
The range of the function ( y = x² + 1 ) is:
[tex] \:\boxed{[1, \infty)}[/tex]
This means ( y ) can be any value starting from 1 and increasing to infinity.
Step-by-step explanation:
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