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Chapter 0: Problem 29

Determine whether the point lies on the graph of the function. $$ P(3,3) ; \quad f(x)=\frac{x+1}{\sqrt{x^{2}+7}}+2 $$

Short Answer

Expert verified
Yes, the point \(P(3,3)\) lies on the graph of the function \(f(x)=\frac{x+1}{\sqrt{x^{2}+7}}+2\). This is because when we substitute the x-coordinate (3) into the function, it simplifies to \(f(3)=3\), which matches the y-coordinate of the point.

Step by step solution

01

Substitute the x-coordinate in the function

Let's replace x with the x-coordinate of the given point (3) in the function and see if it yields the y-coordinate: \[f(3)=\frac{3+1}{\sqrt{3^2+7}}+2\]
02

Simplify the expression

Now, we need to simplify the expression we just got: \[f(3)=\frac{4}{\sqrt{9+7}}+2\] \[f(3)=\frac{4}{\sqrt{16}}+2\] \[f(3)=\frac{4}{4}+2\] \[f(3)=1+2\] \[f(3)=3\]
03

Check if the result equals the y-coordinate of the point

Since after evaluating the function at x=3, we got f(3) = 3, which is equal to the y-coordinate of the given point (3, 3), the point P(3, 3) lies on the graph of the function.

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