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If \[p\left( x \right) = x + 9\], then find the value of \[p\left( x \right) + p\left( { - x} \right)\]?

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Last updated date: 20th Jun 2024
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Answer
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Hint:
Here in this question we will first find the value of \[p\left( { - x} \right)\] by replacing the value of \[x\] by \[ - x\] in the given function. Then we will add the given function and the obtained function to get the required value of \[p\left( x \right) + p\left( { - x} \right)\].

Complete step by step solution:
Given function is \[p\left( x \right) = x + 9\]…………………. \[\left( 1 \right)\]
Now we will find the value of \[p\left( { - x} \right)\] by substituting the value of \[x\] as \[ - x\] in the given function to get the value of \[p\left( { - x} \right)\]. Therefore, we get
 \[ \Rightarrow p\left( { - x} \right) = - x + 9\]…………………. \[\left( 2 \right)\]
Now we will add the equation \[\left( 1 \right)\] and equation \[\left( 2 \right)\] to get the value of \[p\left( x \right) + p\left( { - x} \right)\]. Therefore, we get
 \[p\left( x \right) + p\left( { - x} \right) = \left( {x + 9} \right) + \left( { - x + 9} \right)\]
Opening the bracket, we get
 \[ \Rightarrow p\left( x \right) + p\left( { - x} \right) = x + 9 - x + 9\]
Adding and subtracting the like terms, we get
 \[ \Rightarrow p\left( x \right) + p\left( { - x} \right) = 18\]

Hence the value of \[p\left( x \right) + p\left( { - x} \right)\] is equal to 18.

Additional Information:
Here, the equation given in the question is the linear equation as in this the highest exponent of the variable \[x\] is one. A linear equation has only one solution. For any equation, numbers of roots are always equal to the value of the highest exponent of the variable. Other than linear equations, there are many types of equations such as quadratic equation, cubic equation. Quadratic equation is an equation which has a highest degree of 2. Cubic equation is an equation which has a highest degree of variable as 3.

Note:
Here we might make a mistake by changing the sign of 9 in the equation while finding \[p\left( { - x} \right)\]. Here we just have to replace \[x\] by \[ - x\] and not change the value of the variable and constant. We can also make a mistake that instead of adding \[p\left( { - x} \right)\] and \[p\left( x \right)\] we might subtract them and get the answer as \[2x\], which is incorrect.