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To determine which function has no \( x \)-intercepts, we need to understand what an \( x \)-intercept is. An \( x \)-intercept occurs when the graph of a function crosses the \( x \)-axis, meaning the \( y \)-value is zero at that point.

In mathematical terms, the \( x \)-intercept is found by solving \( f(x) = 0 \) for the given function \( f(x) \). Therefore, a function that **never equals zero** will have no \( x \)-intercepts.

Common examples of functions with **no \( x \)-intercepts** are:
1. **Constant functions** where the output is never zero, e.g., \( f(x) = c \) where \( c \neq 0 \).
2. **Exponential functions**, such as \( f(x) = e^x \), which never touch the \( x \)-axis because \( e^x > 0 \) for all real \( x \).
3. **Square root functions** that have been shifted vertically away from the \( x \)-axis, e.g., \( f(x) = \sqrt{x} + 1 \).
4. **Vertical shifts of certain functions** like \( f(x) = x^2 + 1 \) where the parabola opens upwards and is always positive.

Since the image asks for a multiple-choice selection, I can't see the choices directly. If you'd like, you can provide them, and I can help identify the correct function.

Do you want more details on this, or have any questions?

Here are 5 relative questions to expand on the concept:
1. How do you find \( x \)-intercepts algebraically for a given function?
2. What types of functions are guaranteed to have \( x \)-intercepts?
3. Can a linear function have no \( x \)-intercepts?
4. What is the difference between \( x \)-intercepts and \( y \)-intercepts in a graph?
5. How does the vertical shift of a function affect the \( x \)-intercepts?

**Tip:** Functions involving exponential growth or decay often have no \( x \)-intercepts since their range doesn't include zero.

Which of the following is an example of a function whose graph in the xy-plane has no x-intercepts?

Learn about functions whose graphs have no x-intercepts in the xy-plane. Examples include exponential functions, constant functions, and vertical shifts of parabolas. Suitable for students in Grades 8-10.

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