Solve the equation f(x)=g(x) in f(x)=x^3 -4x and g(x)=3x^2 -12. Interpret the meaning of the solution(s). which of the following statements is correct?

Question

1)the functions f(x) and g(x) intersect when x=-2, x=2, and x=3
2)the functions f(x) and g(x) intersect when x=-3, x=-2, and x=2
3)the functions f(x) and g(x) have x-intercepts at x=-3, x=-2, and x=2
4)the functions f(x) and g(x) have x-intercepts at x=-2, x=2, and x=3

Answer 1

To find the points where the functions \( f(x) = x^3 - 4x \) and \( g(x) = 3x^2 - 12 \) intersect, we need to solve the equation \( f(x) = g(x) \):

\[ x^3 - 4x = 3x^2 - 12. \]

Rearranging this gives:

\[ x^3 - 3x^2 - 4x + 12 = 0. \]

Next, we will attempt to factor the polynomial or use the Rational Root Theorem to find any rational roots. By testing possible rational roots, we can find that \( x = 2 \) is a root:

\[ 2^3 - 3(2^2) - 4(2) + 12 = 8 - 12 - 8 + 12 = 0. \]

Now we can use synthetic division to divide \( x^3 - 3x^2 - 4x + 12 \) by \( x - 2 \):

2 |  1  -3  -4  12
   |      2  -2 -12
--------------------
     1  -1  -6   0

The quotient is \( x^2 - x - 6 \). Next, we can factor \( x^2 - x - 6 \):

\[ x^2 - x - 6 = (x - 3)(x + 2). \]

Thus, we have the complete factorization:

\[ x^3 - 3x^2 - 4x + 12 = (x - 2)(x - 3)(x + 2). \]

Setting the factored polynomial equal to zero gives us the solutions:

\[ x - 2 = 0 \quad \Rightarrow \quad x = 2, \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3, \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2. \]

So, the solutions are \( x = -2, 2, 3 \).

Now, let's analyze the statements:

1. The functions \( f(x) \) and \( g(x) \) intersect when \( x = -2, 2, \) and \( 3 \) - Correct.
2. The functions \( f(x) \) and \( g(x) \) intersect when \( x = -3, -2, \) and \( 2 \) - Incorrect (it includes -3, which is not a solution).
3. The functions \( f(x) \) and \( g(x) \) have x-intercepts at \( x = -3, -2, \) and \( 2 \) - Incorrect (one of the x-intercepts is incorrect).
4. The functions \( f(x) \) and \( g(x) \) have x-intercepts at \( x = -2, 2, \) and \( 3 \) - Incorrect (they are solutions to the intersection, not specifically x-intercepts of the functions themselves).

Thus, the correct statement is:

1) the functions \( f(x) \) and \( g(x) \) intersect when \( x = -2, x = 2, \) and \( x = 3 \).

Interpretation of the Solutions

The solutions \( x = -2, 2, \) and \( 3 \) represent the x-coordinates at which the graphs of the two functions intersect. These points indicate where the outputs of both functions are equal, representing specific inputs for which the results of \( f(x) \) match those of \( g(x) \).