Let’s solve the problem step-by-step.

B.1) Which of the following is a linear function?

A linear function must satisfy the following:

• The highest degree of $x$ and $y$ is 1 (no exponents other than 1 on variables).
• There are no products of variables (like $xy$, $x^2$, etc.).
• The equation must be expressible in the form $y = mx + b$, where $m$ and $b$ are constants.

i) $x^2 + 4y = 13$

• $x^2$ indicates a quadratic term (degree 2), so this is not linear.

ii) $y = \frac{x^4}{3} - 11$

• $\frac{x^4}{3}$ indicates a degree 4 term, so this is not linear.

iii) $x(2 + 8x) = y$

• Expanding $x(2 + 8x)$ gives $2x + 8x^2 = y$, which includes $8x^2$ (degree 2 term). This is not linear.

iv) $y = -7x + 2$

• This is already in the form $y = mx + b$ ($m = -7, b = 2$), so this is linear.

Answer for B.1: iv)

B.2) Which of the following is a nonlinear function?

A nonlinear function violates the rules of a linear function (e.g., involves exponents, products of variables, etc.).

i) $y = x(5 - 9x)$

• Expanding $x(5 - 9x)$ gives $5x - 9x^2$, which includes $x^2$ (degree 2 term). This is nonlinear.

ii) $6x - y = 1$

• Rearranging gives $y = 6x - 1$, which is in the form $y = mx + b$. This is linear.

iii) $y = 3x + 14$

• This is already in the form $y = mx + b$. This is linear.

iv) $x - \frac{4}{5} = y$

• Rearranging gives $y = x - \frac{4}{5}$, which is in the form $y = mx + b$. This is linear.

Answer for B.2: i)

Final Answers:

• B.1: iv) $y = -7x + 2$
• B.2: i) $y = x(5 - 9x)$

Do you have any questions or want additional details? Let me know!

Here are 5 related questions to expand your understanding:

1. How can you determine if an equation is linear just by looking at it?
2. Why is $y = \frac{x^4}{3} - 11$ nonlinear even though it looks "simple"?
3. Can you rewrite $x(5 - 9x) = y$ in standard polynomial form?
4. What happens to a linear equation if you square the variable?
5. How does the slope-intercept form ($y = mx + b$) help identify linear equations?

Tip: A quick way to identify a linear equation is to check if all terms involving variables are of degree 1, with no products of variables or roots.