aphathalo

2022-10-03

Try to determine whether the following function is even or odd: $f(x)=\frac{x}{x+1}$

I pass −x as input to the function to get: $f(-x)=\frac{-x}{-x+1}$

What is next step?

I pass −x as input to the function to get: $f(-x)=\frac{-x}{-x+1}$

What is next step?

Emilia Boyle

Beginner2022-10-04Added 10 answers

By definition, if $f(-x)=f(x)$ then f is even, and if $f(-x)=-f(x)$ then f is odd.

From your calculation, you found: $f(-x)={\displaystyle \frac{-x}{-x+1}}$. This can be rewritten as follows

$$\begin{array}{rl}f(-x)& =\frac{-x}{-x+1}\\ & =\frac{-(-x)}{-(-x+1)}\\ & =\frac{x}{x-1},\end{array}$$

where in the second equality I multiplied both the numerator and the denominator of the fraction by $-1$. So, $f(x)$ is neither even nor odd.

From your calculation, you found: $f(-x)={\displaystyle \frac{-x}{-x+1}}$. This can be rewritten as follows

$$\begin{array}{rl}f(-x)& =\frac{-x}{-x+1}\\ & =\frac{-(-x)}{-(-x+1)}\\ & =\frac{x}{x-1},\end{array}$$

where in the second equality I multiplied both the numerator and the denominator of the fraction by $-1$. So, $f(x)$ is neither even nor odd.

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