Q19. - Algebra 2

Question

State whether each equation or function is linear. Write yes or no. If no, explain your reasoning.

 

$19.g\left(x\right)=10+\frac{2}{x^2}$

Step-by-Step Solution

Answer

No, the function $g\left(x\right)=10+\frac{2}{x^2}$is not a linear function.

Step 1 – Definition of a linear equation and linear function.

In a linear equation the exponent of each variable is 1 and the variables may not be multiplied together. The graph of a linear equation is always a line.

 

In a linear function, each input value has a unique output value. In other words, for each point in domain, there is a unique value in the range of function. It can be written as $f\left(x\right)=mx+b$, where m and b are real numbers.

Step 2 – Given information.

The given function is:

$g\left(x\right)=10+\frac{2}{x^2}$

This function can be rewritten as:

$g\left(x\right)=10+2x^{-2}$                     $\left[\because a^{-n}=\frac{1}{a^n}\right]$

Step 3 – Check whether the equation is linear or not.

In the function $g\left(x\right)=10+2x^{-2}$ the exponent of x is -2.

So, the function $g\left(x\right)=10+\frac{2}{x^2}$ is not a linear function because x has an exponent other than 1.

Thus, the required answer is “No” because the function $g\left(x\right)=10+\frac{2}{x^2}$ is not a linear function.