Find the x and y intercepts of the following function.
f(x) = 9x - x^3
x-intercept: set y or f(x)=0 and solve for x
y-intercept: set x = 0 and solve for y
y-intercept
f(0)=9*0-0^3=0-0=0
x-intercept
9x-x^3=x*(9-x^2)=0
First solution is x=0
Next solution
9-x^2=0
x^2=9
x=+/- sqroot(9)=+/-3
You have 3 solutions
x1=0
X2=-3
x3=3
To find the x-intercepts, we need to find the values of x for which f(x) equals zero. In other words, we need to solve the equation f(x) = 0.
For this particular function, f(x) = 9x - x^3. Setting f(x) = 0, we have:
0 = 9x - x^3
To solve this equation, we can factor out an x:
0 = x(9 - x^2)
Now we have two possible solutions:
1. x = 0 (from the first factor x = 0)
2. 9 - x^2 = 0 (from the second factor)
To find the x-intercepts, we substitute these solutions back into the original equation.
1. When x = 0:
f(0) = 9(0) - (0)^3
f(0) = 0
So, the x-intercept is at x = 0.
2. When 9 - x^2 = 0:
9 - x^2 = 0
x^2 = 9
x = ±√9
x = ±3
So, we have two additional x-intercepts at x = -3 and x = 3.
To find the y-intercept, we substitute x = 0 into the function:
f(0) = 9(0) - (0)^3
f(0) = 0
Therefore, the y-intercept is at y = 0.