# Given `f(x)=2^x` . Determine the value of `(f(x+3)+f(x+1))/(f(x-2))` .

lemjay | High School Teacher | (Level 3) Senior Educator

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To start, f(x+3), f(x+1), and f(x-2) must be determined.

From the given function,

`f(x) = 2^x`

replace the variable x with (x+3) to solve for f(x+3).

`f(x+3) = 2^(x+3)`

To simplify, use the property of exponents for multiplying same base which is `a^(m+n) = a^m*a^n` .

`f(x+3) = 2^x*2^3 = 2^x*8`

>> Hence, `f(x+3) = 8(2^x)` .

For f(x+1), replace the variable x in the given function with x+1.

`f(x+1) = 2^(x+1)`

Again, use the property of exponents for multiplying same base.

`f(x+1) = 2^x*2^1 = 2^x*2 = 2(2^x)`

>> Thus, `f(x+1) = 2(2^x)` .

And for f(x-2), substitute x-2 to `f(x)=2^x` .

`f(x-2) = 2^(x-2) = 2^x*2^(-2)`

Then, apply the negative exponent property which is `a^(-m) = 1/a^m` .

`f(x-2) = 2^x*(1/2^2) = 2^x*(1/4) = 2^x/4`

>> Hence, `f(x-2) = 2^x/4` .

Next, substitute the expression of f(x+3), f(x+1) and f(x-2) to `(f(x+3)+f(x+1))/(f(x-2))` .

`(f(x+3)+f(x+1))/(f(x-2))= (8(2^x)+ 2(2^x))/((2^x)/4)`

At the numerator, factor out `2^x` .

`=[2^x(8+2)]/(2^x/4) = (2^x*10)/(2^x/4)`

To simplify the complex fraction, multiply the top and bottom by 4.

`= (2^x*10) / (2^x/4) * 4/4 = (40*2^x)/2^x`

Then, cancel the `2^x` at the numerator and denominator.

= 40

Hence,  `[f(x+3)+f(x+1)]/[f(x-2)] = 40` .