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f(x)=3^(-2(x+1))
Which of the following equivalent forms of the given function 
f displays, as the base or the coefficient, the 
y-coordinate of the 
y-intercept of the graph of 
y=f(x) in the 
xy-plane?
A) 
f(x)=((1)/(3))^((2x+2))
B) 
f(x)=(1)/(9)((1)/(3))^(2x)
C) 
f(x)=81^((-(1)/(2)x-(1)/(2)))
D) 
f(x)=3^((-2x-2))

f(x)=32(x+1) f(x)=3^{-2(x+1)} \newlineWhich of the following equivalent forms of the given function f f displays, as the base or the coefficient, the y y -coordinate of the y y -intercept of the graph of y=f(x) y=f(x) in the xy x y -plane?\newlineA) f(x)=(13)(2x+2) f(x)=\left(\frac{1}{3}\right)^{(2 x+2)} \newlineB) f(x)=19(13)2x f(x)=\frac{1}{9}\left(\frac{1}{3}\right)^{2 x} \newlineC) f(x)=81(12x12) f(x)=81^{\left(-\frac{1}{2} x-\frac{1}{2}\right)} \newlineD) f(x)=3(2x2) f(x)=3^{(-2 x-2)}

Full solution

Q. f(x)=32(x+1) f(x)=3^{-2(x+1)} \newlineWhich of the following equivalent forms of the given function f f displays, as the base or the coefficient, the y y -coordinate of the y y -intercept of the graph of y=f(x) y=f(x) in the xy x y -plane?\newlineA) f(x)=(13)(2x+2) f(x)=\left(\frac{1}{3}\right)^{(2 x+2)} \newlineB) f(x)=19(13)2x f(x)=\frac{1}{9}\left(\frac{1}{3}\right)^{2 x} \newlineC) f(x)=81(12x12) f(x)=81^{\left(-\frac{1}{2} x-\frac{1}{2}\right)} \newlineD) f(x)=3(2x2) f(x)=3^{(-2 x-2)}
  1. Evaluate f(x)f(x) at x=0x=0: To find the y-intercept of the graph of y=f(x)y = f(x), we need to evaluate f(x)f(x) when x=0x = 0.
  2. Substitute x=0x=0 into f(x)f(x): Substitute x=0x = 0 into the function f(x)=32(x+1)f(x) = 3^{-2(x+1)}.f(0)=32(0+1)=32=132=19f(0) = 3^{-2(0+1)} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}. The yy-coordinate of the yy-intercept is 19\frac{1}{9}.
  3. Check options for y-intercept: Now we need to find which of the given options has the y-coordinate of the y-intercept 19\frac{1}{9} as the base or the coefficient.
  4. Option A analysis: Option A: f(x)=(13)(2x+2)f(x) = \left(\frac{1}{3}\right)^{(2x+2)} can be rewritten as f(x)=(13)2x+2f(x) = \left(\frac{1}{3}\right)^{2x+2}. This does not have 19\frac{1}{9} as the base or the coefficient.
  5. Option B analysis: Option B: f(x)=19(13)2xf(x) = \frac{1}{9}\left(\frac{1}{3}\right)^{2x} has the coefficient 19\frac{1}{9}, which is the y-coordinate of the y-intercept.
  6. Option C analysis: Option C: f(x)=81(12x12)f(x) = 81^{(-\frac{1}{2}x-\frac{1}{2})} can be rewritten as f(x)=(34)12x12=32x2f(x) = (3^4)^{-\frac{1}{2}x-\frac{1}{2}} = 3^{-2x-2}, which does not have 19\frac{1}{9} as the base or the coefficient.
  7. Option D analysis: Option D: f(x)=3(2x2)f(x) = 3^{(-2x-2)} is the same as the original function and does not have 19\frac{1}{9} as the base or the coefficient.

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