Answer :
To determine whether the function [tex]\( y = x(x + 1) \)[/tex] is linear or nonlinear, let's analyze the structure of this function.
1. Understand the Expression: The function given is [tex]\( y = x(x + 1) \)[/tex].
2. Expand the Function:
- Start by distributing the [tex]\( x \)[/tex] across the terms in the parentheses:
[tex]\[
y = x \cdot x + x \cdot 1
\][/tex]
- This simplifies to:
[tex]\[
y = x^2 + x
\][/tex]
3. Identify the Type of Function:
- A linear function is one that can be written in the form [tex]\( y = ax + b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and the highest power of [tex]\( x \)[/tex] is 1.
- A nonlinear function involves powers of [tex]\( x \)[/tex] other than 1, such as [tex]\( x^2 \)[/tex], [tex]\( x^3 \)[/tex], etc., or other operations, like roots or trigonometric functions.
4. Check the Highest Power of [tex]\( x \)[/tex]:
- In the expanded function [tex]\( y = x^2 + x \)[/tex], the highest power of [tex]\( x \)[/tex] is 2.
5. Conclusion:
- Since the function [tex]\( y = x^2 + x \)[/tex] involves [tex]\( x^2 \)[/tex], it is a quadratic function.
- Quadratic functions, such as this one, are types of nonlinear functions because they do not conform to the structure of a linear function (which must have [tex]\( x \)[/tex] raised only to the power of 1).
Therefore, the function [tex]\( y = x(x + 1) \)[/tex] is nonlinear.
1. Understand the Expression: The function given is [tex]\( y = x(x + 1) \)[/tex].
2. Expand the Function:
- Start by distributing the [tex]\( x \)[/tex] across the terms in the parentheses:
[tex]\[
y = x \cdot x + x \cdot 1
\][/tex]
- This simplifies to:
[tex]\[
y = x^2 + x
\][/tex]
3. Identify the Type of Function:
- A linear function is one that can be written in the form [tex]\( y = ax + b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and the highest power of [tex]\( x \)[/tex] is 1.
- A nonlinear function involves powers of [tex]\( x \)[/tex] other than 1, such as [tex]\( x^2 \)[/tex], [tex]\( x^3 \)[/tex], etc., or other operations, like roots or trigonometric functions.
4. Check the Highest Power of [tex]\( x \)[/tex]:
- In the expanded function [tex]\( y = x^2 + x \)[/tex], the highest power of [tex]\( x \)[/tex] is 2.
5. Conclusion:
- Since the function [tex]\( y = x^2 + x \)[/tex] involves [tex]\( x^2 \)[/tex], it is a quadratic function.
- Quadratic functions, such as this one, are types of nonlinear functions because they do not conform to the structure of a linear function (which must have [tex]\( x \)[/tex] raised only to the power of 1).
Therefore, the function [tex]\( y = x(x + 1) \)[/tex] is nonlinear.
