Math Problem Statement
Solution
The given function is:
To find the x-intercept and y-intercept, we follow these steps:
1. Finding the y-intercept:
The y-intercept occurs when . Substitute into the equation and solve for :
[ y = 2 \sqrt[3]{0 - 8} + 2 = 2 \sqrt[3]{-8} + 2 = 2(-2) + 2 = -4 + 2 = -2 ]
Thus, the y-intercept is at .
2. Finding the x-intercept:
The x-intercept occurs when . Set and solve for :
First, subtract 2 from both sides:
Now, divide by 2:
Next, cube both sides to eliminate the cube root:
Solve for :
Thus, the x-intercept is at .
Summary:
- x-intercept:
- y-intercept:
Would you like more details or have any questions about this solution?
Here are 5 related questions you could explore:
- How would the intercepts change if the function had a different constant term?
- What is the domain of the function ?
- How do you find the asymptotes for functions with cube roots?
- Can this function have multiple x-intercepts?
- How would the graph of the function look, and how would you plot it manually?
Tip: When solving for intercepts, always remember that the y-intercept involves setting , while the x-intercept involves setting .
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Math Problem Analysis
Mathematical Concepts
Algebra
Cube Roots
Intercepts
Formulas
y-intercept occurs at x = 0
x-intercept occurs at y = 0
Solving cube root equations: ∛a
Theorems
Intercept theorem: y-intercept and x-intercept definitions
Suitable Grade Level
Grades 10-12