Consider the functions f and g given by f x( ) = −log 110 ( )x x+ +log 310 ( ) and g x( ) = +log 910 ( )x . In thexy-plane, what are all x -coordinates of the points of intersection of the graphs of f and g ?(A) x = 3 only(B) x = 7(C) x = −4 and x = 3(D) x = − 7 and x = −4
Question
Consider the functions f and g given by f x( ) = −log 110 ( )x x+ +log 310 ( ) and g x( ) = +log 910 ( )x . In thexy-plane, what are all x -coordinates of the points of intersection of the graphs of f and g ?(A) x = 3 only(B) x = 7(C) x = −4 and x = 3(D) x = − 7 and x = −4
Solution
The points of intersection of the graphs of f and g are the solutions to the equation f(x) = g(x).
This means we need to solve the equation:
−log10(x) + log10(x+3) = log10(9x)
We can simplify this equation using the properties of logarithms.
First, remember that log(a) - log(b) = log(a/b). So, we can rewrite the left side of the equation as:
log10[(x+3)/x] = log10(9x)
Next, remember that if log(a) = log(b), then a = b. So, we can simplify the equation to:
(x+3)/x = 9x
Solving this equation for x gives us the x-coordinates of the points of intersection of the graphs of f and g.
Multiply both sides by x to get rid of the denominator on the left side:
x+3 = 9x^2
Rearrange the equation to set it equal to zero:
9x^2 - x - 3 = 0
This is a quadratic equation, which we can solve using the quadratic formula:
x = [-(-1) ± sqrt((-1)^2 - 49(-3))]/(2*9)
Solving this gives us two solutions: x = -1/3 and x = 1.
So, the x-coordinates of the points of intersection of the graphs of f and g are x = -1/3 and x = 1.
However, these solutions are not in the provided options, which suggests that there may be a mistake in the problem or in my calculations. Please check the problem and the calculations.
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