Solve for x in the equation f(x) = 4x + 1 when f(x) = 13.
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Let's solve for x in the equation f(x) = 4x + 1 when f(x) = 13. We start by setting the given value of f(x) equal to the expression for f(x). So we have 13 equals 4x plus 1. Next, we subtract 1 from both sides of the equation. 13 minus 1 equals 4x plus 1 minus 1, which simplifies to 12 equals 4x. Finally, we divide both sides by 4 to solve for x. 12 divided by 4 equals 4x divided by 4, which gives us x equals 3. Therefore, the solution is x equals 3. We can verify this by substituting x equals 3 back into the original function: f(3) = 4(3) + 1 = 12 + 1 = 13.
Now, let's verify our solution by substituting x equals 3 back into the original function. We have f of 3 equals 4 times 3 plus 1. Computing this, we get f of 3 equals 12 plus 1, which equals 13. Since we were told that f of x equals 13, and we've shown that f of 3 equals 13, our solution x equals 3 is indeed correct! We can visualize this on our graph by looking at the point where the function f of x equals 4x plus 1 intersects with the horizontal line y equals 13. This intersection occurs exactly at the point (3, 13), confirming our algebraic solution.
Now, let's explore what happens when we change the value of f(x) in our equation. For Case 1, if f(x) equals 5, we set 5 equal to 4x plus 1. Subtracting 1 from both sides, we get 4 equals 4x. Dividing both sides by 4, we find that x equals 1. For Case 2, if f(x) equals 9, we set 9 equal to 4x plus 1. Subtracting 1 from both sides, we get 8 equals 4x. Dividing both sides by 4, we find that x equals 2. We can visualize these different cases on our graph. As we move the horizontal line up and down to different y-values, the intersection point with our function changes, giving us different solutions for x. For any value of f(x), we can solve for x using the same algebraic steps we used earlier.