For this type of question, do you need a graph to figure out the solution or is there another non graphical way to know which is the top and bottom function? I used desmos to graph the functions but how would you do this question without looking at desmos?

Step-by-step explanation:

First, find the intersections.

4x = x³

0 = x³ − 4x

0 = x (x² − 4)

0 = x (x + 2) (x − 2)

x = -2, 0, 2

So we will consider two intervals: -2 ≤ x ≤ 0, and 0 ≤ x ≤ 2.

Pick a point within each interval and evaluate both functions at that point.  Compare to see which function is greater.

At x = -1, y₁ = 4x = -4, and y₂ = x³ = -1.  So y₂ > y₁.

At x = 1, y₁ = 4x = 4, and y₂ = x³ = 1.  So y₁ > y₂.

So the integral would be:

∫₋₂⁰ (x³ − 4x) dx + ∫₀² (4x − x³) dx

f(x) = x^3 -2x^2 -3x

x( x^2 -2x-3)

x( x^2 +x -3x -3)

x( x(x+1) -3(x+1))

x( x-3)( x+1)

so zeroes are 0, 3,-1

its (2) option

step-by-step explanation:

f(x)=2x^2 -1 

translation is a function transformation that transform the function in a way of translating. by translating the function its graph is moved  up an down. moving the function f(x) 2 units to the left will be the new function g(x) with the equation: g(x)=f(x+2)=2(x+2)^2 -1