To determine which of the given functions are nonlinear, let's analyze each option one by one. A function is considered nonlinear if its graph does not produce a straight line. This means the variables are involved in operations other than addition and subtraction, such as multiplications involving variables, exponents higher than 1, divisions involving variables, logarithms, etc.
1. Option A: [tex]\(9y + 3 = 0\)[/tex]
- This equation can be rewritten in the form [tex]\(y = -\frac{1}{3}\)[/tex], which is a linear equation since it represents a horizontal line. There are no higher-order terms.
2. Option B: [tex]\(y - 4x = 1\)[/tex]
- Rearranging gives [tex]\(y = 4x + 1\)[/tex]. This is a linear equation in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. The relation between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is linear.
3. Option C: [tex]\(y = 2 + 6x^4\)[/tex]
- This equation includes a term [tex]\(6x^4\)[/tex], which is a fourth power of [tex]\(x\)[/tex]. Because of this higher-order exponent, it is not linear. The presence of [tex]\(x\)[/tex] raised to the power of 4 makes it a nonlinear equation.
4. Option D: [tex]\(x - 2y = 7\)[/tex]
- Rearranging gives [tex]\(x = 2y + 7\)[/tex]. This is also a linear equation in the form [tex]\(x = my + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the intercept. The relation is linear.
5. Option E: [tex]\(\frac{x}{y} + 1 = 2\)[/tex]
- Simplifying gives [tex]\(\frac{x}{y} = 1\)[/tex], which implies [tex]\(x = y\)[/tex]. This equation involves the division of variables. The relationship [tex]\(x = y\)[/tex] can be solved directly, but the initial relation involving a division indicates nonlinearity.
Conclusion:
Options C and E are nonlinear equations. Hence, the correct answers are:
- Option C: [tex]\(y = 2 + 6x^4\)[/tex]
- Option E: [tex]\(\frac{x}{y} + 1 = 2\)[/tex]
