First slide

Functions (XII)

Question

$If f (x) = 5 \log_{5} x then f^{- 1} (α - β) where α, β \in R is equal to$

Moderate

A

$f^{- 1} (α) - f^{- 1} (β)$
B

$\frac{f^{- 1} (α)}{f^{- 1} (β)}$
C

$\frac{1}{f (α - β)}$
D

$\frac{1}{f (α) - f (β)}$

Solution

$l e t f (x) = 5 \log {}_{5}x = y \Rightarrow \log {}_{5}x = \frac{y}{5} \Rightarrow x = 5^{\frac{y}{5}}$
$t h e r e f o r e f^{- 1} (x) = 5^{\frac{x}{5}} \Rightarrow f^{- 1} (α - β) = 5^{\frac{(α - β)}{5}} = \frac{5^{\frac{α}{5}}}{5^{\frac{β}{5}}} = \frac{f^{- 1} (α)}{f^{- 1} (β)}$

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