Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?

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To find the inverse of the function \(f(x) = -x + 1\), start by replacing \(f(x)\) with \(y\): write \(y = -x + 1\).

Next, swap the variables \(x\) and \(y\) to find the inverse function: write \(x = -y + 1\).

Now, solve this equation for \(y\) to express the inverse function: rearrange to get \(y\) in terms of \(x\).

For graphing, plot the original line \(y = -x + 1\) and the line \(y = x\) on the same coordinate plane. These lines will intersect at a point which you can find by setting \(-x + 1 = x\) and solving for \(x\).

To find the angle of intersection between the two lines, use the formula for the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\): \(\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\). Here, identify the slopes of the lines and substitute them into this formula.

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주요 개념

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Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For f(x) = -x + 1, the inverse is found by solving y = -x + 1 for x and then expressing y in terms of x. The inverse function reflects the original function across the line y = x.

Graphing Linear Functions

Graphing linear functions involves plotting lines based on their slope and y-intercept. The line y = -x + 1 has a slope of -1 and intercept 1, while y = x has slope 1 and intercept 0. Visualizing both lines helps understand their relationship and points of intersection.

Angle Between Two Lines

The angle between two intersecting lines can be found using their slopes. If m1 and m2 are slopes, the angle θ satisfies tan(θ) = |(m2 - m1) / (1 + m1*m2)|. This formula calculates the acute angle where the lines cross, important for understanding their geometric relationship.