Let be a real number such that and

MATHEMATICS · Subjective · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

Let

a

be a real number such that

a>0

and

a\neq1

. Define

h(x)=\log_{a}(x+2)+5

. (i) Prove that

h

is invertible on its domain. (ii) Obtain

h^{-1}(x)

in terms of

a

. (iii) State the exact domain and range of

h^{-1}

. (iv) If

a<1

, explain how the monotonic nature of

h

and

h^{-1}

differs from the case

a>1