Question

Find an expression for a cubic function f if \(f\left( 1 \right) = 6\) and

\(f\left( { - 1} \right) = f\left( 0 \right) = f\left( 2 \right) = 0\)

Short answer

The expression of the cubic function is \( - 3{x^3} + 3x + 6x\)

Step-by-step solution

Cubic function

A cubic function is afunction that has one or more variables, but in which the highest exponent of the variable is three.

Finding the expression of the cubic function

It is given that

\(f\left( 1 \right) = 6\)

And

\(f\left( { - 1} \right) = f\left( 0 \right) = f\left( 2 \right) = 0\)

The function is zero at\(x = - 1\),\(x = 0\)and\(x = 2\)

Therefore, these are three roots of the cubic function.

Hence, we can express the function as

\(\begin{array}{l}f\left( x \right) = a\left( {x - \left( { - 1} \right)} \right)\left( {x - 0} \right)\left( {x - 2} \right)\\f\left( x \right) = ax\left( {x + 1} \right)\left( {x - 2} \right)\end{array}\)

Where a is constant, to find the value of a.

We know,\(f\left( 1 \right) = 6\)

Substituting the value into the expression

\(\begin{array}{c}f\left( 1 \right) = a\left( 1 \right)\left( {1 + 1} \right)\left( {1 - 2} \right)\\6 = a\left( 2 \right)\left( { - 1} \right)\\a = - 3\end{array}\)

Substituting the value of a into the function

\(\begin{array}{c}f\left( x \right) = - 3x\left( {x + 1} \right)\left( {x - 2} \right)\\ = - 3x\left( {{x^2} - 2x - x - 2} \right)\\ = - 3{x^3} + 3x + 6x\end{array}\)

The expression of the cubic function is \(\left( { - 3{x^3} + 3x + 6x} \right)\)