From health tips to tech hacks, find it all on IDNLearn.com. Our experts provide timely and precise responses to help you understand and solve any issue you face.
Sagot :
Let's break down the function [tex]\( y = (x + 3)^2 - 5 \)[/tex] to understand its domain and range:
### Domain
The domain of a function refers to all the possible values of [tex]\(x\)[/tex] that can be input into the function.
In the function [tex]\( y = (x + 3)^2 - 5 \)[/tex], there are no restrictions on [tex]\(x\)[/tex]. This is because the term [tex]\((x + 3)^2\)[/tex] is a polynomial expression, which can handle any real number input. Therefore, [tex]\(x\)[/tex] can be any real number.
So, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range
The range of a function refers to all the possible values that [tex]\(y\)[/tex] can take as [tex]\(x\)[/tex] varies over the domain.
The function [tex]\( y = (x + 3)^2 - 5 \)[/tex] is in the form of a quadratic function, which represents a parabola. The standard form of a quadratic function is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola and the parabola opens upwards if [tex]\(a > 0\)[/tex] and downwards if [tex]\(a < 0\)[/tex].
For [tex]\( y = (x + 3)^2 - 5 \)[/tex]:
- The parabola opens upwards because the coefficient of the squared term [tex]\((x + 3)^2\)[/tex] is positive (1).
- The vertex of the parabola can be obtained directly from the function. Here, [tex]\( (h, k) \)[/tex] is [tex]\((-3, -5)\)[/tex].
As the parabola opens upwards, the smallest value of [tex]\(y\)[/tex] is at the vertex, which corresponds to [tex]\( y = -5 \)[/tex]. Therefore, the minimum value of [tex]\( y \)[/tex] is [tex]\(-5\)[/tex], and as [tex]\(x\)[/tex] moves away from the vertex in either direction, [tex]\( y \)[/tex] increases without bound.
Thus, the range of the function is:
[tex]\[ [-5, \infty) \][/tex]
Matching our findings with the provided options:
- The domain is [tex]\( (-\infty, \infty) \)[/tex].
- The range is [tex]\( [-5, \infty) \)[/tex].
Therefore, the correct answer is:
D. Domain: [tex]\( (-\infty, \infty) \)[/tex]
Range: [tex]\([-5, \infty)\)[/tex]
### Domain
The domain of a function refers to all the possible values of [tex]\(x\)[/tex] that can be input into the function.
In the function [tex]\( y = (x + 3)^2 - 5 \)[/tex], there are no restrictions on [tex]\(x\)[/tex]. This is because the term [tex]\((x + 3)^2\)[/tex] is a polynomial expression, which can handle any real number input. Therefore, [tex]\(x\)[/tex] can be any real number.
So, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range
The range of a function refers to all the possible values that [tex]\(y\)[/tex] can take as [tex]\(x\)[/tex] varies over the domain.
The function [tex]\( y = (x + 3)^2 - 5 \)[/tex] is in the form of a quadratic function, which represents a parabola. The standard form of a quadratic function is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola and the parabola opens upwards if [tex]\(a > 0\)[/tex] and downwards if [tex]\(a < 0\)[/tex].
For [tex]\( y = (x + 3)^2 - 5 \)[/tex]:
- The parabola opens upwards because the coefficient of the squared term [tex]\((x + 3)^2\)[/tex] is positive (1).
- The vertex of the parabola can be obtained directly from the function. Here, [tex]\( (h, k) \)[/tex] is [tex]\((-3, -5)\)[/tex].
As the parabola opens upwards, the smallest value of [tex]\(y\)[/tex] is at the vertex, which corresponds to [tex]\( y = -5 \)[/tex]. Therefore, the minimum value of [tex]\( y \)[/tex] is [tex]\(-5\)[/tex], and as [tex]\(x\)[/tex] moves away from the vertex in either direction, [tex]\( y \)[/tex] increases without bound.
Thus, the range of the function is:
[tex]\[ [-5, \infty) \][/tex]
Matching our findings with the provided options:
- The domain is [tex]\( (-\infty, \infty) \)[/tex].
- The range is [tex]\( [-5, \infty) \)[/tex].
Therefore, the correct answer is:
D. Domain: [tex]\( (-\infty, \infty) \)[/tex]
Range: [tex]\([-5, \infty)\)[/tex]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.