Determine The \[$x\$\]- And \[$y\$\]-intercepts For The Given Function. Write Your Answer As An Ordered Pair.$\[ R(x) = \sqrt{x} - 7 \\]If There Is More Than One Answer, Separate Your Answers With Commas. Select None If

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Understanding the Problem

To determine the x- and y-intercepts for the given function, we need to understand what these intercepts represent. The x-intercept is the point where the graph of the function crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.

The Given Function

The given function is:

r(x)=x−7{ r(x) = \sqrt{x} - 7 }

This function represents a square root function with a constant term subtracted from it.

Finding the x-Intercept

To find the x-intercept, we need to set the function equal to zero and solve for x.

x−7=0{ \sqrt{x} - 7 = 0 }

We can add 7 to both sides of the equation to isolate the square root term.

x=7{ \sqrt{x} = 7 }

Now, we can square both sides of the equation to eliminate the square root.

x=72{ x = 7^2 }

x=49{ x = 49 }

Therefore, the x-intercept is (49, 0).

Finding the y-Intercept

To find the y-intercept, we need to set x equal to zero and solve for y.

r(0)=0−7{ r(0) = \sqrt{0} - 7 }

r(0)=0−7{ r(0) = 0 - 7 }

r(0)=−7{ r(0) = -7 }

Therefore, the y-intercept is (0, -7).

Conclusion

In conclusion, the x-intercept for the given function is (49, 0), and the y-intercept is (0, -7).

Key Takeaways

  • The x-intercept is the point where the graph of the function crosses the x-axis.
  • The y-intercept is the point where the graph crosses the y-axis.
  • To find the x-intercept, set the function equal to zero and solve for x.
  • To find the y-intercept, set x equal to zero and solve for y.

Practice Problems

  1. Find the x- and y-intercepts for the function:

f(x)=2x−5{ f(x) = 2x - 5 }

  1. Find the x- and y-intercepts for the function:

g(x)=x+3{ g(x) = \sqrt{x} + 3 }

Answer Key

  1. The x-intercept is (-2.5, 0), and the y-intercept is (0, -5).
  2. The x-intercept is (9, 0), and the y-intercept is (0, 3).

Additional Resources

For more information on finding x- and y-intercepts, check out the following resources:

  • Khan Academy: Finding x- and y-intercepts
  • Mathway: Finding x- and y-intercepts
  • Wolfram Alpha: Finding x- and y-intercepts
    Frequently Asked Questions (FAQs) about Finding x- and y-Intercepts ====================================================================

Q: What is the x-intercept?

A: The x-intercept is the point where the graph of a function crosses the x-axis. It is the value of x where the function equals zero.

Q: What is the y-intercept?

A: The y-intercept is the point where the graph of a function crosses the y-axis. It is the value of y where the function equals zero.

Q: How do I find the x-intercept of a function?

A: To find the x-intercept of a function, set the function equal to zero and solve for x. This will give you the value of x where the function crosses the x-axis.

Q: How do I find the y-intercept of a function?

A: To find the y-intercept of a function, set x equal to zero and solve for y. This will give you the value of y where the function crosses the y-axis.

Q: What if the function has multiple x-intercepts?

A: If the function has multiple x-intercepts, you will need to solve the equation for x multiple times. Each solution will give you a different x-intercept.

Q: What if the function has no x-intercepts?

A: If the function has no x-intercepts, it means that the function does not cross the x-axis. This can happen if the function is always positive or always negative.

Q: Can I use a graphing calculator to find x- and y-intercepts?

A: Yes, you can use a graphing calculator to find x- and y-intercepts. Graph the function and use the calculator to find the points where the graph crosses the x- and y-axes.

Q: Are x- and y-intercepts the same as the zeros of a function?

A: Yes, x- and y-intercepts are the same as the zeros of a function. The zeros of a function are the values of x where the function equals zero.

Q: Can I find x- and y-intercepts for a function with a variable in the denominator?

A: No, you cannot find x- and y-intercepts for a function with a variable in the denominator. This is because the function may not be defined for all values of x.

Q: Can I find x- and y-intercepts for a function with a square root in the numerator?

A: Yes, you can find x- and y-intercepts for a function with a square root in the numerator. However, you will need to be careful when solving the equation, as the square root may introduce extraneous solutions.

Q: Can I find x- and y-intercepts for a function with a logarithm in the numerator?

A: Yes, you can find x- and y-intercepts for a function with a logarithm in the numerator. However, you will need to be careful when solving the equation, as the logarithm may introduce extraneous solutions.

Q: Can I find x- and y-intercepts for a function with a trigonometric function in the numerator?

A: Yes, you can find x- and y-intercepts for a function with a trigonometric function in the numerator. However, you will need to be careful when solving the equation, as the trigonometric function may introduce extraneous solutions.

Conclusion

In conclusion, finding x- and y-intercepts is an important concept in algebra and calculus. By understanding how to find x- and y-intercepts, you can better understand the behavior of functions and make predictions about their behavior.