Consider the functions 𝑓 and 𝑔 given by 𝑓⁡(𝑥)=log10⁡(𝑥-1)+log10⁡(𝑥+3) and 𝑔⁡(𝑥)=log10⁡(𝑥+9). In the 𝑥⁢𝑦-plane, what are all 𝑥-coordinates of the points of intersection of the graphs of 𝑓 and 𝑔 ?

To find the x-coordinates of the points of intersection of the graphs of f and g, we need to set the two functions equal to each other and solve for x.

So, we have:

log10(x-1) + log10(x+3) = log10(x+9)

Using the properties of logarithms, we can combine the left side of the equation:

log10[(x-1)(x+3)] = log10(x+9)

This simplifies to:

(x-1)(x+3) = x+9

Expanding and simplifying gives us a quadratic equation:

x^2 + 2x - 3 = x + 9

Rearranging terms gives:

x^2 + x - 12 = 0

This factors to:

(x - 3)(x + 4) = 0

Setting each factor equal to zero gives the solutions x = 3 and x = -4. However, we must check these solutions in the original equation because logarithms are undefined for negative numbers and zero.

For x = 3, we have log10(3-1) + log10(3+3) = log10(3+9), which simplifies to log10(2) + log10(6) = log10(12). This is a true statement, so x = 3 is a solution.

For x = -4, we have log10(-4-1) + log10(-4+3) = log10(-4+9), which simplifies to log10(-5) + log10(-1) = log10(5). However, logarithms are undefined for negative numbers, so x = -4 is not a solution.

Therefore, the only x-coordinate of the point of intersection of the graphs of f and g is x = 3.