Question

if f(x)=a^x show that f(x-h)=f(x)/f(h) Question by relation and function

Answer 1

$\underline{ \underline{\large{ \pink{\bf{given}}}}}$

$\longrightarrow\sf{f(x) = {a}^{x} }$

$\underline{ \underline{ \large{ \bf{ \pink{to \: prove}}}}}$

$\longrightarrow \sf{f(x - h) = \dfrac{f(x)}{f(h)} }$

$\underline{ \underline{ \large{ \bf{ \pink{proof}}}}}$

As given

$\implies \sf{f(x) = {a}^{x} }$

Replacing (x) by (x - h)

$\implies \sf{f(x - h) = {a}^{(x - h)} }$

using rule of indices

$\boxed{\sf{a^{(m-n)}=\dfrac{a^m}{a^n}} }$

$\implies \sf{f(x - h) = \dfrac{ {a}^{x} }{ {a}^{h} } }$

using the given Info

$\implies \sf{f(x - h) = \dfrac{ f(x) }{ f(h)}}$

( where f(h) ≠ 0 )

Proved.

Additional information related to functions :

• Let A and B be two non empty sets. then a function f from set A to set B is a method or correspondence which associates elements of set A to elements of set B such that: 1) all elements of set A are associated to elements in set B. 2) an element of set A is associated to a unique element in set B.

• Let f be a function from set A to B then, the set A is known as the domain of function f and the set B is known as the co- domain of f . the set of all f-images of elements of A is known as the range of function f or image of set of A under function f and is denoted by f (A).

Answer 2

given

• $\rm{f(x) = {a}^{x} }$

To find

• show that $\rm{f(x - h) = \dfrac{f(x)}{f(h)} }$

Solution

ATQ

$\implies \rm{f(x) = {a}^{x}}$

Replacing (x) by (x - h)

$\implies \rm{f(x - h) = {a}^{(x - h)} }$

using rule of indices

$\boxed{\rm{a^{(m-n)}=\dfrac{a^m}{a^n}} }$

$\implies \rm{f(x - h) = \dfrac{ {a}^{x} }{ {a}^{h} } }$

using the given Info

$\implies \boxed {\boxed {\rm{f(x - h) = \dfrac{ f(x) }{ f(h)}}}}$