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

How do you find g(x+1)g(x+1) if we are given g(x)=x3−2x2g(x)={{x}^{3}}-2{{x}^{2}}?

Explanation

Solution

As we have been given a value of a function g(x)=x3−2x2g(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)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)=x3−2x2g(x)={{x}^{3}}-2{{x}^{2}}.
We have to find the value of g(x+1)g(x+1).
As the given g(x)=x3−2x2g(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)g(x+1) let us replace the x by x+1 in the given equation. Then we will get
⇒g(x+1)=(x+1)3−2(x+1)2\Rightarrow g(x+1)={{\left( x+1 \right)}^{3}}-2{{\left( x+1 \right)}^{2}}
Now, we have an algebraic identities as (a+b)3=(a+b)(a2+b2+ab){{\left( a+b \right)}^{3}}=\left( a+b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right) and (a+b)2=(a2+b2+2ab){{\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
⇒g(x+1)=(x+1)(x2+12+2x)−2(x2+12+2x)\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
⇒g(x+1)=(x+1)(x2+1+2x)−2(x2+1+2x)\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
⇒g(x+1)=(x2+1+2x)(x+1−2)\Rightarrow g(x+1)=\left( {{x}^{2}}+1+2x \right)\left( x+1-2 \right)
Now, simplifying the above obtained equation we will get
⇒g(x+1)=(x2+1+2x)(x−1)\Rightarrow g(x+1)=\left( {{x}^{2}}+1+2x \right)\left( x-1 \right)
Hence we get the value of g(x+1)=(x2+1+2x)(x−1)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)=(x+1)2(x−1)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 ⇒g(x+1)=(x2+1+2x)(x+1)−2\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.