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The graph of y=g(x+5) in the xy-plane passes through the point (-5,4). If g is an exponential function, which of the following could define g ? Choose 1 answer: (A) g(x)=-5(4)^(x)+1 (B) g(x)=4^(x) (c) g(x)=4((4)/(5))^(x) (D) g(x)=4(2)^(x)-5

The graph of y=g(x+5) y=g(x+5) in the xy x y -plane passes through the point (−5,4) (-5,4) . If g g is an exponential function, which of the following could define g g ?\newlineChoose 11 answer:\newline(A) g(x)=−5(4)x+1 g(x)=-5(4)^{x}+1 \newline(B) g(x)=4x g(x)=4^{x} \newline(C) g(x)=4(45)x g(x)=4\left(\frac{4}{5}\right)^{x} \newline(D) g(x)=4(2)x−5 g(x)=4(2)^{x}-5

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Full solution

Q. The graph of y=g(x+5) y=g(x+5) in the xy x y -plane passes through the point (−5,4) (-5,4) . If g g is an exponential function, which of the following could define g g ?\newlineChoose 11 answer:\newline(A) g(x)=−5(4)x+1 g(x)=-5(4)^{x}+1 \newline(B) g(x)=4x g(x)=4^{x} \newline(C) g(x)=4(45)x g(x)=4\left(\frac{4}{5}\right)^{x} \newline(D) g(x)=4(2)x−5 g(x)=4(2)^{x}-5
  1. Substitute and Solve: To find the correct function gg, we need to substitute the given point (−5,4)(-5,4) into the equation y=g(x+5)y=g(x+5) and solve for gg. Since the point (−5,4)(-5,4) lies on the graph, when x=−5x=-5, yy should equal 44.
  2. Check Options: Substituting x=−5x=-5 into the equation y=g(x+5)y=g(x+5), we get y=g(0)y=g(0). Since y=4y=4 when x=−5x=-5, we have g(0)=4g(0)=4. This means that the value of the function gg at x=0x=0 must be 44.
  3. Option (A): Now we will check each option to see which function satisfies g(0)=4g(0)=4.\newline(A) g(x)=−5(4)x+1g(x)=-5(4)^{x}+1\newlineg(0)=−5(4)0+1=−5(1)+1=−5+1=−4g(0)=-5(4)^{0}+1=-5(1)+1=-5+1=-4, which does not equal 44.
  4. Option (B): g(x)=4xg(x)=4^{x}g(0)=40=1g(0)=4^{0}=1, which does not equal 44.
  5. Option (C): g(x)=4(45)xg(x)=4\left(\frac{4}{5}\right)^{x}g(0)=4(45)0=4(1)=4g(0)=4\left(\frac{4}{5}\right)^{0}=4(1)=4, which equals 44. This function satisfies the condition g(0)=4g(0)=4.
  6. Option (D): g(x)=4(2)x−5g(x)=4(2)^{x}-5g(0)=4(2)0−5=4(1)−5=4−5=−1g(0)=4(2)^{0}-5=4(1)-5=4-5=-1, which does not equal 44.

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