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The graphs of a function $$f(x)=3x+b$$ and its inverse function $$f^{-1}(x)$$ intersect at the point $$(-3,a)$$. Given that $$b$$ and $$a$$ are both integers, what is the value of $$a$$?

Feb 17, 2019

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$$\text{if }v=f(u) \text{ then } u = f^{-1}(v) \text{ in other words}\\ \text{if }(u,v) \text{ is a point on }f(u) \text{ then }(v,u) \text{ must be a point on }f^{-1}(v)\\ \text{thus if these two points are identical}\\ (u,v)=(v,u) \Rightarrow u=v$$

$$\text{so if }(-3,a) \text{ is on the graph of }f(x)\\ (a,-3) \text{ is on the graph of }f^{-1}(x) \\ \text{and if they coincide then }a=-3$$

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Feb 17, 2019