How are the graphs #f(x)=sqrtx# and #g(x)=-sqrt(0.4x)+3# related?

#g(x)# is an adjusted (transformation) of #f(x)#

Consider these points individually
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#color(blue)("x->0.4x)#

The multiply by #0.4# 'shifts' any point on #sqrt(x)# to the left
by #sqrt(x-0.6x)# in that #1-0.4=0.6#

In other words it changes the horizontal scale of the plot
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#color(blue)(+3)#

This raises the #sqrt(x)# by 3 so if #y=sqrt(x)# then #y+3=sqrt(x)+3#
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#color(blue)("Negating "sqrt(x)" "->" "-sqrt(x))#

For this condition it would reflect #+sqrt(x)# about the x-axis in that what was a positive #y# value becomes a negative #y# value.

However we have lifted the whole thing by the addition of 3 so it will in fact be reflected (rotated) about a line parallel to the x-axis.

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#color(blue)("The line if reflection")#

If we did not have the + 3 it would be the x-axis. However, by adding 3 we have effectively lifted the line of reflection as well as the graphed line by 3.

Thus as this is not a #ul("direct reflection")# of the original image should we call it such? #color(brown)("It is more like a transformation")#

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#color(blue)("Determine the point of intersection.")#

Equate #f(x)=g(x)# to find #x_("intersection")#

#sqrt(x)=-sqrt(0.4x)+3#

Changed my mind about how to resolve this.

Write as: #" "sqrt(x)=-sqrt(0.4)sqrt(x)+3#

#sqrt(x)(1+sqrt(0.4))=3#

#x=(3/(1+sqrt(0.4)))^2 = 3.377223..... #

#color(red)("I will let you solve the rest" ) #

Substitute the value of #x# in #f(x)# to determine #y_("intersection")# and thus the line if reflection.

Notice that the curve near the y axis of the inverted plot (red) is visibly compressed when compared to the other plot (blue). The other point differences are harder to observe but they are there.