Multiple Choice
Find the domain and range of the following graph (write your answer using interval notation).
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3. Functions
Intro to Functions & Their Graphs
3:03 minutesProblem 7a
Textbook Question
Find the domain of each function. g(x) = 3/(x2-2x-15)
Verified step by step guidance1
Step 1: Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like g(x) = 3/(x^2 - 2x - 15), the function is undefined when the denominator equals zero.
Step 2: Set the denominator equal to zero to find the x-values that make the function undefined. Solve the equation x^2 - 2x - 15 = 0.
Step 3: Factor the quadratic expression x^2 - 2x - 15. Look for two numbers that multiply to -15 and add to -2. These numbers are -5 and 3, so the factorization is (x - 5)(x + 3).
Step 4: Solve the factored equation (x - 5)(x + 3) = 0. Set each factor equal to zero: x - 5 = 0 and x + 3 = 0. Solve for x to find x = 5 and x = -3.
Step 5: Exclude the values x = 5 and x = -3 from the domain, as these make the denominator zero. The domain of g(x) is all real numbers except x = 5 and x = -3, which can be written in interval notation as (-∞, -3) ∪ (-3, 5) ∪ (5, ∞).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically restricted by values that would make the denominator zero, as division by zero is undefined.
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Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting them as a product of their linear factors. This is essential for identifying the values that make the denominator zero. In the case of g(x) = 3/(x^2-2x-15), factoring the quadratic will help determine the critical points that affect the domain.
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Finding Asymptotes
Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur at the x-values that make the denominator zero. Understanding asymptotes is crucial for determining the domain, as these points indicate where the function is undefined.
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