Question
Give me 10 different equations of a graph that have no solutions
Answers
GPT-4o mini
Here are ten different equations of graphs that have no solutions, meaning they do not intersect the x-axis (i.e., they do not have any real roots):
1. **Quadratic Equation**: \( y = x^2 + 1 \)
- This graph is a parabola that opens upwards and is always above the x-axis.
2. **Linear Equation**: \( y = 2x + 3 \)
- This linear equation has a y-intercept of 3 and a positive slope, so it is always above the x-axis.
3. **Exponential Equation**: \( y = e^x + 2 \)
- The exponential function \( e^x \) is always positive, so adding 2 keeps the entire expression above zero.
4. **Logarithmic Equation**: \( y = \ln(x) + 5 \) (for \( x > 0 \))
- The logarithmic function \( \ln(x) \) is undefined for \( x \leq 0 \) and becomes positive for \( x > 0 \) when adding 5.
5. **Cubic Equation**: \( y = x^3 + 1 \)
- This cubic function will never touch the x-axis as it trends to positive infinity.
6. **Absolute Value Equation**: \( y = |x| + 4 \)
- The absolute value function is always non-negative, and adding 4 ensures it is always above the x-axis.
7. **Sine Function**: \( y = \sin(x) + 1 \)
- The sine function oscillates between -1 and 1; adding 1 shifts it above the x-axis.
8. **Cosine Function**: \( y = \cos(x) + 2 \)
- Similar to the sine function, the cosine function oscillates between -1 and 1, so this graph is always above the x-axis.
9. **Rational Function**: \( y = \frac{1}{x} + 3 \) (for \( x \neq 0 \))
- For \( x > 0 \), \( \frac{1}{x} \) is positive, and adding 3 ensures the graph remains above the x-axis. For \( x < 0 \), the expression remains negative.
10. **Gaussian Function**: \( y = e^{-x^2} + 1 \)
- The Gaussian function \( e^{-x^2} \) reaches a maximum of 1 when \( x = 0 \), so the entire expression is always positive.
These equations represent graphical functions that do not intersect the x-axis, indicating that there are no solutions where \( y = 0 \).
1. **Quadratic Equation**: \( y = x^2 + 1 \)
- This graph is a parabola that opens upwards and is always above the x-axis.
2. **Linear Equation**: \( y = 2x + 3 \)
- This linear equation has a y-intercept of 3 and a positive slope, so it is always above the x-axis.
3. **Exponential Equation**: \( y = e^x + 2 \)
- The exponential function \( e^x \) is always positive, so adding 2 keeps the entire expression above zero.
4. **Logarithmic Equation**: \( y = \ln(x) + 5 \) (for \( x > 0 \))
- The logarithmic function \( \ln(x) \) is undefined for \( x \leq 0 \) and becomes positive for \( x > 0 \) when adding 5.
5. **Cubic Equation**: \( y = x^3 + 1 \)
- This cubic function will never touch the x-axis as it trends to positive infinity.
6. **Absolute Value Equation**: \( y = |x| + 4 \)
- The absolute value function is always non-negative, and adding 4 ensures it is always above the x-axis.
7. **Sine Function**: \( y = \sin(x) + 1 \)
- The sine function oscillates between -1 and 1; adding 1 shifts it above the x-axis.
8. **Cosine Function**: \( y = \cos(x) + 2 \)
- Similar to the sine function, the cosine function oscillates between -1 and 1, so this graph is always above the x-axis.
9. **Rational Function**: \( y = \frac{1}{x} + 3 \) (for \( x \neq 0 \))
- For \( x > 0 \), \( \frac{1}{x} \) is positive, and adding 3 ensures the graph remains above the x-axis. For \( x < 0 \), the expression remains negative.
10. **Gaussian Function**: \( y = e^{-x^2} + 1 \)
- The Gaussian function \( e^{-x^2} \) reaches a maximum of 1 when \( x = 0 \), so the entire expression is always positive.
These equations represent graphical functions that do not intersect the x-axis, indicating that there are no solutions where \( y = 0 \).