How do you graph, find the intercepts and state the domain and range of #f(x)=2-2^x#?
1 Answer
Jun 8, 2018
Refer points below.
Explanation:
- We have to find out the Zeroes of the
#f(x)# .
#f(x)=2-2^x=0#
#=># #2^x=2#
#=># #x=1#
We have one point on the graph is#(1,0)# - First Derivative Test:
We have find first derivative of#f(x)# .
#f(x)=2-2^x#
#f'(x)=-2^xln2#
Note that#f'(x)# is always negative as#2^x>0# and#ln2>0# .
#f'(x)<0# for#AA x inRR#
#=># #f(x)# is always deceasing.-------(By First Derivative Test). - Second Derivative Test:
We have,
#f(x)=2-2^x#
#f'(x)=-2^xln2#
#f''(x)=-2^x(ln2)^2#
Note that#f''(x)# is always negative.
#f''(x)<0# for#AA x inRR#
#=># #f(x)# is always concave downward.-------(By second Derivative Test). - Points on co-ordinate axes:
We have already a point on#x# -axis i.e.#(1,0)# (as stated in point No. 1).
We have to find out a point on#y# -axis.
So, put#x=0# in#f(x)#
#f(x)=2-2^x#
#:.# #f(0)=2-2^0#
#:.# #f(0)=2-1#
#:.# #f(0)=1# - Domain:
Clearly there is not any point for which#f(x)# is not defined.
Hence,#D_f=(-oo,oo)# - Range:
#f(x)# cannot exceed the#y=2# . Also,#f(x)# cannot achieve#y=2#
Hence,#R_f=(-oo,2)# - Intercepts:
#x# -intercept= 1 unit
#y# -intercept= 1 unit
We have found it by point No.4. - Graph:
By above Points we are able to draw the graph.
graph{2-2^x [-10, 10, -5, 5]}
