Given #h(x)=e^-x#, how do you describe the transformation?

Answer 1

It is a reflection across the y-axis of #f(x)=e^x#

I'm going to assume you mean the transformation from the parent function #f(x) = e^x#.
Let's think about how to get #e^-x# from #f(x)#:
You would have to plug in #-x# instead of #x#, giving #f(-x)=e^-x#

Therefore we can say that:

#h(x) = f(-x)#

What does this mean as far as transformation on a graph? Well...

The point #(2, e^2)# on #f(x)# will correspond to the point #(-2, e^2)# on #h(x)#.

In fact, plugging in any value into f(x) and then plugging in the negative of that value to h(x) will give the same answer.

This basically means that the x-values of #f(x)# become the negative of what they were originally.
Graphically, this means the graph of #f(x)# is reflected across the y-axis to get #h(x)#.
You can see this below: #e^x# and #e^-x# are graphed together to show how #e^-x# is a reflection of #e^x#.
#color(white)"XXXXXXXXXXXXXX-"e^-x color(white)"XXXXX"e^x# #color(white)"XXXXXXXXXXXXXX-"darr color(white)"XXXXX"darr# graph{(y-e^x)(y-e^-x)=0 [-10.16, 9.84, -3.4, 6.6]}
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Answer 2

The function ( h(x) = e^{-x} ) represents an exponential function with a base of ( e ) raised to the power of ( -x ). The transformation of this function compared to the parent function ( e^x ) involves a reflection across the x-axis. This reflection flips the graph of the function vertically, causing it to decrease rapidly as ( x ) increases. There are no horizontal or vertical shifts or stretches/compressions involved in this transformation.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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