$\begin{aligned} y \geq x^2+6 \quad \text { use } 0 & =x \\ 0 & =y \end{aligned}$ What is $X$-intercept? $(C, F)$ What is $y$-intercept? $(x ; x)$
Answer (1)
To find the x-intercept, set y = 0 in the inequality, resulting in x 2 l e − 6 . Since this is impossible for real numbers, there is no x-intercept.
To find the y-intercept, set x = 0 in the inequality, resulting in y g e 6 . The y-intercept is ( 0 , 6 ) .
There is no x-intercept.
The y-intercept is ( 0 , 6 ) .
Explanation
Understanding the Problem We are given the inequality y g e x 2 + 6 . We need to find the x-intercept (C, F) and the y-intercept (x, x). We are also given that we should use x = 0 and y = 0 .
Finding the X-Intercept To find the x-intercept, we set y = 0 in the inequality y g e x 2 + 6 . This gives us 0 g e x 2 + 6 , which can be rewritten as x 2 + 6 l e 0 or x 2 l e − 6 . Since x 2 is always non-negative for real numbers, x 2 cannot be less than or equal to − 6 . Therefore, there are no real x-intercepts.
Finding the Y-Intercept To find the y-intercept, we set x = 0 in the inequality y g e x 2 + 6 . This gives us y g e 0 2 + 6 , which simplifies to y g e 6 . This means the y-intercept occurs when x = 0 and y = 6 . So, the y-intercept is ( 0 , 6 ) .
Final Answer Since there is no x-intercept, we can say there is no solution for (C, F). The y-intercept is ( 0 , 6 ) , so ( x , x ) = ( 0 , 6 ) .
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, the x-intercept of a cost function could represent the break-even point, while the y-intercept represents the fixed costs. Similarly, in physics, the y-intercept of a velocity-time graph indicates the initial velocity of an object. By finding intercepts, we can gain valuable insights into the behavior of functions and their practical implications.
