Question

Express the relationship f(x) is𝛉g((x)) using a picture. Show that graphs of the function f(x), C₁│g(x)│ , and C₂│g(x)│, as well as the constant k on the x-axis.

Short answer

Using the graph we expressed the relationship f(x) is θ(g(x))

Step-by-step solution

Step 1

Here f(x) isθ(g(x))

By the definition of Big-theta notation,

we can say f(x) is O(g(x)) an f(x) is Ω(g(x))

let C₁ be the constant for Big-O notation and C₂ be the constant for Big-Omega notation.

Step 2

If x is sufficiently large, then f(x) is bounded and f(x) cannot be closer to 0.

Whenever x is large the function C₂│g(x))│is between the function f(x) and the x-axis as f(x) cannot loser to 0

And C₂│g(x))│ is further from the horizontal axis than the function f(x) [since f(x) is bounded] The constant k is the right point of intersection between f(x) and two curves C₁│g(x))│and C₂│g(x))│

Final answer

Using the graph we expressed the relationship f(x) is θ(g(x))