Question

Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ f(x)=\frac{1}{x} ; \quad g(x)=\frac{1}{x} $$

Short answer

We found that \(f[g(x)] = f[\frac{1}{x}] = \frac{1}{\frac{1}{x}} = x\) and \(g[f(x)] = g[\frac{1}{x}] = \frac{1}{\frac{1}{x}} = x\). Since both expressions simplify to \(x\), the functions \(f(x)\) and \(g(x)\) are inverses of each other.

Step-by-step solution

Find f[g(x)]

To find \(f[g(x)]\), we need to substitute \(g(x)\) into \(f(x)\). So, we get: \(f[g(x)] = f[\frac{1}{x}]\)

Substitute g(x) into f(x)

Now that we have our expression for \(f[g(x)]\), we can substitute the function \(g(x)\) into \(f(x)\). This substitution will give us the following: \(f[g(x)] = \frac{1}{\frac{1}{x}}\)

Simplify f[g(x)]

Now we will simplify the expression to see if it is equal to \(x\): \(f[g(x)] = \frac{x}{1}\) So, we find that \(f[g(x)] = x\). This verifies the first condition for inverse functions.

Find g[f(x)]

Next, we need to find \(g[f(x)]\) by substituting \(f(x)\) into \(g(x)\): \(g[f(x)] = g[\frac{1}{x}]\)

Substitute f(x) into g(x)

Like with step 2, now we will substitute the function \(f(x)\) into \(g(x)\) to get the following expression: \(g[f(x)] = \frac{1}{\frac{1}{x}}\)

Simplify g[f(x)]

To check if \(g[f(x)]\) is equal to \(x\), we will simplify the expression: \(g[f(x)] = \frac{x}{1}\) Now we find that \(g[f(x)] = x\). This verifies the second condition for inverse functions.

Conclusion

Since we have verified that \(f[g(x)] = x\) and \(g[f(x)] = x\), we can conclude that the functions \(f(x)\) and \(g(x)\) are inverses of each other.