How do you graph, find any intercepts, domain and range of f(x)=-(3/4)^(x+2)+5f(x)=−(34)x+2+5?

1 Answer
Somebody N.
Apr 8, 2018

See below.

Explanation:

f(x)=-(3/4)^(x+2)+5f(x)=−(34)x+2+5

yy axis intercepts occur where x=0x=0

y=-(3/4)^(0+2)+5=-(3/4)^2+5=-9/16+5=color(blue)(71/16)y=−(34)0+2+5=−(34)2+5=−916+5=7116

xx axis intercepts occur where y=0y=0

-(3/4)^(x+2)+5=0−(34)x+2+5=0

(3/4)^(x+2)=5(34)x+2=5

Taking logarithms of both sides:

(x+2)ln(3/4)=ln(5)(x+2)ln(34)=ln(5)

x=color(blue)((ln(5))/(ln(3/4))-2)x=ln(5)ln(34)−2

There are no restriction on xx, so the domain is:

{x in RR}

To find the range we need to see what happens as x approaches +-oo

as x->oo, \ \ \ \ -(3/4)^(x+2)+5 -> 5

as x->-oo, \ \ \ \ -(3/4)^(x+2)+5 -> -oo

So the range is:

{y in RR : 5 < y < oo}

The graph confirms these findings:

enter image source here

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