If `g(y)=y^(2)-(1)/(y+1)`, what is `g((1)/(x))`?

Answer

To find \( g\left(\frac{1}{x}\right) \) for the function \( g(y) = y^2 - \frac{1}{y + 1} \), we will follow these steps:

### Step 1: Substitute \( y \) with \( \frac{1}{x} \)

We start by substituting \( y \) in the function \( g(y) \) with \( \frac{1}{x} \):

\[ g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 - \frac{1}{\frac{1}{x} + 1} \]

### Step 2: Simplify \( \left(\frac{1}{x}\right)^2 \)

Calculating \( \left(\frac{1}{x}\right)^2 \):

\[ \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \]

### Step 3: Simplify the second term \( \frac{1}{\frac{1}{x} + 1} \)

Now we simplify \( \frac{1}{\frac{1}{x} + 1} \):

\[ \frac{1}{\frac{1}{x} + 1} = \frac{1}{\frac{1 + x}{x}} = \frac{x}{1 + x} \]

### Step 4: Combine the terms

Now we combine the two parts we have:

\[ g\left(\frac{1}{x}\right) = \frac{1}{x^2} - \frac{x}{1 + x} \]

### Step 5: Find a common denominator

To combine the fractions, we need a common denominator. The common denominator will be \( x^2(1 + x) \):

\[ g\left(\frac{1}{x}\right) = \frac{1(1 + x) - x^3}{x^2(1 + x)} \]

### Step 6: Simplify the numerator

Now we simplify the numerator:

\[ 1(1 + x) - x^3 = 1 + x - x^3 \]

### Step 7: Write the final expression

Thus, we have:

\[ g\left(\frac{1}{x}\right) = \frac{1 + x - x^3}{x^2(1 + x)} \]

### Final Answer

The final expression for \( g\left(\frac{1}{x}\right) \) is:

\[ g\left(\frac{1}{x}\right) = \frac{1 + x - x^3}{x^2(1 + x)} \]

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