How do you find a formula of a function given the function f(x)=e^x ?

Question

Given the function f(x)=e^x

A) Find a formula for g(x) where the graph of g(x) is obtained from the graph of f(x) by shifting up 5 units and then reflecting about the x axis.

B) Find a formula for h(x) where the graph of h(x) is obtained from the graph of f(x) by reflecting about the x-axis and then shifting up 5 units.

C) Are the functions g(x) and h(x) the same? If not, how are their graphs related?

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Answer 1 · 2018-06-12T21:34:34

#g(x) = -e^x-5#
#h(x) -e^x+5#
C) #g(x)# and #h(x)# are translated version of the same function.

In general, given a function #f(x)#, you:

As you can see, the transformations are not commutative:

Shift and then reflect:

#f(x) \to f(x)+k \to -(f(x)+k) = -f(x)-k#

Reflect and then shift:

#f(x) \to -f(x)\to -f(x)+k#

So, in your case, A) leads to

#g(x) = -e^x-5#

while B) leads to

#h(x) = -e^x+5#

which means that A) is the function #-e^x# translated #5# units down, while B) is the function #-e^x# translated #5# units up.

This means that the two functions are the same graph, translated at #10# units of distance.