Given the function f (x) = log4 (4 ^ x + 1), let H (x) = log4 (a * 2 ^ x-3a / 4), if the image of function f (x) and H (x) has and only has one common point Find the value range of a

Given the function f (x) = log4 (4 ^ x + 1), let H (x) = log4 (a * 2 ^ x-3a / 4), if the image of function f (x) and H (x) has and only has one common point Find the value range of a

If f (x) = g (x) has only one solution, then
4^x+1=a*2^x-3a/4
Let 2 ^ x = t
t^2+1-at-3a/4=0
Δ=a^2-4(1-3a/4)=0
The solution is A1 = - 1, A2 = 4
When A1 = - 1, the solution is t = - 1 / 20
When A2 = 4, the solution is t = 2 and x = 1
So a = 4

Find the range of function y = 2x-5 ＋ 15-4x. The range is {y ∣ y ≤ 2.5})

y=2x－5＋√15－4x
=1/2(4x-15)+2.5＋√15－4x
Let z = √ 15-4x
Then z > = 0
y=-1/2z^2 +z+2.5
=-1/2(z-1)^2+3
So when z = 1, i.e. = √ 15-4x = 1, y has a maximum value of 3
So y

The function values of y = x square + 2x-3 form the set a
Want to know how to get here!

Y = x square + 2x-3 = (x + 1) square-4, when x = - 1, there is a minimum value of y = - 4;
So set a = {y | Y > = - 4}