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Questions and Answers
What are the x-intercepts and y-intercept of the function f(x) = √(25 - x^2)?
The x-intercepts are (-5, 0) and (5, 0), and the y-intercept is (0, 5).
How does the value of 'a' in the function f(x) = √(a^2 - x^2) affect the graph of the function?
The value of 'a' affects the width of the graph. A larger value of 'a' results in a wider graph, while a smaller value of 'a' results in a narrower graph.
Explain the domain of the function f(x) = √(25 - x^2). Why is the domain restricted to certain values of x?
The domain of the function f(x) = √(25 - x^2) is x ≤ 5 and x ≥ -5. This restriction is because the square root function requires a non-negative value inside the radical, hence the domain is restricted to ensure that the expression inside the square root is non-negative.
Explain the process of determining the domain of the function f(x) = √(a^2 - x^2).
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Why is it important to be comfortable with the graph of f(x) = √(a^2 - x^2) in calculus?
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Study Notes
Function f(x) = √(25 - x^2)
- The x-intercepts of the function f(x) = √(25 - x^2) are -5 and 5, as these are the values of x that make the function equal to zero.
- The y-intercept of the function f(x) = √(25 - x^2) is 5, as this is the value of the function when x is 0.
Effect of 'a' on the Graph of f(x) = √(a^2 - x^2)
- The value of 'a' in the function f(x) = √(a^2 - x^2) affects the graph of the function by changing its height and width: a larger 'a' results in a taller and wider graph, while a smaller 'a' results in a shorter and narrower graph.
Domain of f(x) = √(25 - x^2)
- The domain of the function f(x) = √(25 - x^2) is restricted to the values of x between -5 and 5, inclusive, because the square root of a negative number is not a real number.
- The domain restriction is due to the fact that the expression inside the square root must be non-negative, which is only true when x is between -5 and 5.
Determining the Domain of f(x) = √(a^2 - x^2)
- To determine the domain of the function f(x) = √(a^2 - x^2), set the expression inside the square root greater than or equal to zero and solve for x, which will give the range of values of x for which the function is defined.
Importance of f(x) = √(a^2 - x^2) in Calculus
- Being comfortable with the graph of f(x) = √(a^2 - x^2) is important in calculus because it is a fundamental function that appears in many calculus problems, including optimization and integration problems.
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