Math Problem Statement
Solution
Let's analyze the image step by step and solve the questions regarding the function .
Questions from the image:
(a) Is the point on the graph of ?
We need to substitute into the function and check if the result equals 1.
Yes, the point is on the graph of .
(b) If , what is ? What point is on the graph of ?
Substitute into the function: So, the point on the graph when is .
(c) If , what is ? What point(s) are on the graph of ?
Set and solve for : Multiply both sides by : Rearrange the equation: Thus: The points on the graph are and .
(d) What is the domain of ?
Since the function is a rational function with no restrictions (no values that make the denominator zero), the domain is all real numbers:
(e) List the x-intercepts, if any, of the graph of .
To find the x-intercepts, set : This implies that , which results in . Thus, the x-intercept is .
(f) List the y-intercept, if there is one, of the graph of .
The y-intercept occurs when : Thus, the y-intercept is .
Do you need more details on any part of this explanation?
5 Related Questions:
- How do we find vertical and horizontal asymptotes of this rational function?
- Can you explain how to calculate the derivative of to find critical points?
- How does the behavior of the function change as approaches infinity or negative infinity?
- What is the range of the function ?
- How would you graph the function and its key points?
Tip:
For rational functions, always check for asymptotes by analyzing the denominator and limits of the function as approaches large values.
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Math Problem Analysis
Mathematical Concepts
Algebra
Functions
Rational Functions
Intercepts
Domain
Formulas
f(x) = (6x^2) / (x^2 + 9)
x-intercepts: set f(x) = 0
y-intercepts: evaluate f(0)
Solving f(x) = 1
Theorems
Rational function domain and intercept properties
Algebraic manipulation to solve for x
Suitable Grade Level
Grades 9-12
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