The functions f(x)=8((2)/(5))^(x) and g(x)=8(b)^(x) are graphed in the xy-plane. For what value of b would the graphs of functions f and g be symmetric with respect to the y-axis?

Question

The functions  f(x)=8((2)/(5))^(x) and  g(x)=8(b)^(x) are graphed in the  xy-plane. For what value of  b would the graphs of functions  f and  g be symmetric with respect to the  y-axis?

Bytelearn · Accepted Answer

Find b for symmetry: We need to find the value of b that makes the function g(x) = 8b^x symmetric to f(x) = 8(2/5)^x with respect to the y-axis. Symmetry with respect to the y-axis means that f(x) = f(-x) for the function f. Since g(x) is supposed to be symmetric to f(x), we need to find b such that g(x) = g(-x).Express g(-x): Let's first express g(-x) using the function g(x) = 8b^x. g(-x) = 8b^(-x) = 8/b^xSet g(x) = g(-x): Now, we want g(x) to be equal to g(-x) for symmetry with respect to the y-axis. So, we set 8b^x equal to 8/b^x and solve for b. 8b^x = 8/b^xSolve for b: To solve for b, we can multiply both sides by b^x to get rid of the denominator on the right side. b^x * 8b^x = b^x * 8/b^x 8b^(2x) = 8Isolate b^(2x): Now, we divide both sides by 8 to isolate b^(2x). b^(2x) = 1Determine x: Since any non-zero number raised to the power of 0 is 1, we have: 2x = 0Consider positive b: Solving for x, we get: x = 0Final value of b: However, we are not looking for the value of x; we are looking for the value of b. Since b^(2x) = 1 for all x, b must be a number that when squared gives 1. The only real numbers that satisfy this condition are 1 and -1. Since we are looking for symmetry with respect to the y-axis, we need to consider positive values of b.Therefore, the value of b that makes g(x) symmetric to f(x) with respect to the y-axis is b = 1.

The functions $f(x)=8\left(\frac{2}{5}\right)^{x}$ and $g(x)=8(b)^{x}$ are graphed in the $x y$ -plane. For what value of $b$ would the graphs of functions $f$ and $g$ be symmetric with respect to the $y$ -axis?

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The functions f(x)=8(25)x f(x)=8\left(\frac{2}{5}\right)^{x} f(x)=8(52​)x and g(x)=8(b)x g(x)=8(b)^{x} g(x)=8(b)x are graphed in the xy x y xy-plane. For what value of b b b would the graphs of functions f f f and g g g be symmetric with respect to the y y y-axis?

The functions $f(x)=8\left(\frac{2}{5}\right)^{x}$ and $g(x)=8(b)^{x}$ are graphed in the $x y$ -plane. For what value of $b$ would the graphs of functions $f$ and $g$ be symmetric with respect to the $y$ -axis?