Question

if ( f(x)=x + 5 ), what is ( f^{-1}(x) )?
( (x + 5)^2 )
( (x + 5)^{-1} )
( x - 5 )
undefined

Explanation:

Step1: Recall the formula for the derivative of a linear function

The derivative of \(y = ax + b\) is \(y'=a\). For \(f(x)=x + 5\), where \(a = 1\) and \(b=5\).

Step2: Apply the formula

Using the formula \(f'(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\), substitute \(f(x)=x + 5\) and \(f(x + h)=(x + h)+5\). Then \(f'(x)=\lim_{h
ightarrow0}\frac{(x + h+5)-(x + 5)}{h}=\lim_{h
ightarrow0}\frac{h}{h}\). Since \(h
eq0\) when taking the limit (\(h
ightarrow0\) but \(h
eq0\)), \(\lim_{h
ightarrow0}\frac{h}{h}=1\).

Answer:

1 (Note: None of the given options \((x + 5)^2\), \((x + 5)^{-1}\), \(x - 5\), undefined are correct. The derivative of \(f(x)=x + 5\) is \(1\))