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 need to find the value of x that makes the function equal to 13.
First, we graph the function f(x) = 4x + 1. This is a linear function with slope 4 and y-intercept 1.
Next, we draw a horizontal line at y = 13, representing f(x) = 13. The x-coordinate of the intersection point is our solution.
To solve algebraically, we set the given value of f(x) equal to the expression: 13 = 4x + 1.
We subtract 1 from both sides: 13 minus 1 equals 4x.
This simplifies to 12 equals 4x.
Dividing both sides by 4, we get x equals 3.
Therefore, the solution is x equals 3. Let's verify this on our graph.
When x equals 3, f of x equals 4 times 3 plus 1, which is 12 plus 1, which equals 13. So our solution is correct.
Now, let's verify our solution. We found that x equals 3. Let's substitute this value back into the original function.
Here's our function f(x) = 4x + 1 again. We need to check if f(3) equals 13.
Substituting x equals 3 into the function, we get f(3) = 4 times 3 plus 1.
This simplifies to f(3) = 12 plus 1.
Which equals 13. So f(3) = 13, which matches our requirement. This confirms that x = 3 is indeed the correct solution.
On our graph, we can see that when x equals 3, the function value is 13. The vertical line at x equals 3 intersects with the function graph at the point where y equals 13.
This intersection point (3, 13) is our solution. The x-coordinate 3 is the value we were looking for.
Therefore, x equals 3 is the solution to the equation f(x) = 4x + 1 when f(x) = 13. Both our algebraic solution and graphical verification confirm this result.
Now, let's explore what happens when we change the value of f(x). We'll look at different cases and find the corresponding x values.
Here's our function f(x) = 4x + 1 again. We'll start with our original case where f(x) = 13, which gave us x = 3.
Now, let's change the value of f(x) to 9 and see what happens.
For Case 1, when f(x) = 9, we solve the equation 9 = 4x + 1. Subtracting 1 from both sides gives us 8 = 4x, so x = 2.
Let's try another case. What if f(x) = 5?
For Case 2, when f(x) = 5, we solve 5 = 4x + 1. This gives us 4 = 4x, so x = 1.
Finally, let's see what happens when f(x) = 1.
For Case 3, when f(x) = 1, we have 1 = 4x + 1. Subtracting 1 from both sides gives us 0 = 4x, so x = 0.
Do you notice the pattern? As the value of f(x) decreases, the corresponding x value also decreases. This makes sense because our function has a positive slope of 4.
For any value of f(x), we can find the corresponding x by using the formula x = (f(x) - 1) divided by 4, which comes from solving the equation f(x) = 4x + 1 for x.