Question

How do you find $g(x+1)$ if we are given $g(x)={{x}^{3}}-2{{x}^{2}}$?

Answer 1

Hint: As we have been given a value of a function $g(x)={{x}^{3}}-2{{x}^{2}}$. The given is the function of x and returns a value in the form of x. So to find the value of $g(x+1)$ we will put the value x+1 in the given expression at place of x and simplify the obtained expression to get the desired answer.

Complete step-by-step solution:
We have been given that $g(x)={{x}^{3}}-2{{x}^{2}}$.
We have to find the value of $g(x+1)$.
As the given $g(x)={{x}^{3}}-2{{x}^{2}}$ is the function of x and gives the value in the form of x.
So to find the value of $g(x+1)$ let us replace the x by x+1 in the given equation. Then we will get
$\Rightarrow g(x+1)={{\left( x+1 \right)}^{3}}-2{{\left( x+1 \right)}^{2}}$
Now, we have an algebraic identities as ${{\left( a+b \right)}^{3}}=\left( a+b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right)$ and ${{\left( a+b \right)}^{2}}=\left( {{a}^{2}}+{{b}^{2}}+2ab \right)$.
Now, applying both the identities in the above obtained equation we will get
$\Rightarrow g(x+1)=\left( x+1 \right)\left( {{x}^{2}}+{{1}^{2}}+2x \right)-2\left( {{x}^{2}}+{{1}^{2}}+2x \right)$
Now, simplifying the above obtained equation we will get
$\Rightarrow g(x+1)=\left( x+1 \right)\left( {{x}^{2}}+1+2x \right)-2\left( {{x}^{2}}+1+2x \right)$
Now, taking common terms out we will get
$\Rightarrow g(x+1)=\left( {{x}^{2}}+1+2x \right)\left( x+1-2 \right)$
Now, simplifying the above obtained equation we will get
$\Rightarrow g(x+1)=\left( {{x}^{2}}+1+2x \right)\left( x-1 \right)$
Hence we get the value of $g(x+1)=\left( {{x}^{2}}+1+2x \right)\left( x-1 \right)$.

Note: We can further simplify the above obtained equation as $g(x+1)={{\left( x+1 \right)}^{2}}\left( x-1 \right)$. Students can make a mistake while taking the terms common. They can write the terms as \[\Rightarrow g(x+1)=\left( {{x}^{2}}+1+2x \right)\left( x+1 \right)-2\]. After simplifying the obtained equation we will get the incorrect solution. So be careful while taking the common terms out and solve the question step by step to avoid mistakes.