determine if the function shown on the graph is continuous. if not, identify the type (infinite, jump, or…
determine if the function shown on the graph is continuous. if not, identify the type (infinite, jump, or removable) and location of discontinuity.\n1.\ncontinuous\ndiscontinuous\ntype: ____________\nlocation: ____________\nuse the graph to determine if the relation is symmetrical to the x - axis, y - axis, and/or origin.\n1. $y = x^{4}-4x^{2}$\nx - axis\ny - axis\norigin\nnone\ndescribe the end behavior of the graph.\n5.
Answer
Explanation:
Step1: Check continuity for first - graph
A function is continuous if there are no breaks, jumps, or holes. The first graph has a hole, so it is discontinuous.
Step2: Identify type of discontinuity
A removable discontinuity is a hole in the graph. Since the first graph has a hole, it is a removable discontinuity.
Step3: Locate discontinuity
The hole in the first graph is at the x - value where the open - circle is located. By observing the graph, it is at (x = 3).
Step4: Check symmetry for (y=x^{4}-4x^{2})
Replace (x) with (-x): (y=(-x)^{4}-4(-x)^{2}=x^{4}-4x^{2}). Since (y(-x)=y(x)), the function is symmetric about the y - axis. If it were symmetric about the x - axis, replacing (y) with (-y) would give the original equation, which is not the case. If it were symmetric about the origin, replacing (x) with (-x) and (y) with (-y) would give the original equation, which is not true here.
Step5: Analyze end - behavior for second graph
As (x\to+\infty), the function value (y\to0) (the graph approaches the x - axis from above). As (x\to-\infty), the function value (y\to0) (the graph approaches the x - axis from above).
Answer:
- Discontinuous Type: Removable Location: (x = 3)
- y - axis
- As (x\to+\infty), (y\to0); as (x\to-\infty), (y\to0)