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Express the function graphed on the axes below as a piecewise function.

Express the function graphed on the axes below as a piecewise function.-example-1
User Plancke
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9 votes

Answer:

f(x) = {x-3 for x ≤ -1; -3x+14 for x > 5}

Explanation:

To write the piecewise function, we can consider the pieces one at a time. For each, we need to define the domain, and the functional relation.

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Left Piece

The domain is the horizontal extent. It is shown as -∞ to -1, with -1 included.

The relation has a slope (rise/run) of +1, and would intersect the y-axis at -3 if it were extended.

The first piece can be written ...

f(x) = x-3 for x ≤ -1

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Right Piece

The domain is shown as 5 to ∞, with 5 not included.

The relation is shown as having a slope (rise/run) of (-3)/(1) = -3. If extended, it would intersect the point (5, -1), so we can write the point-slope equation as ...

y -(-1) = -3(x -5)

y = -3x +15 -1 = -3x +14

The second piece can be written ...

f(x) = -3x +14 for x > 5

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Whole function

Putting these pieces together, we have ...


\boxed{f(x)=\begin{cases}x-3&amp;\text{for }x\le-1\\-3x+14&amp;\text{for }5 < x\end{cases}}

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Additional comment

Sometimes it is convenient to write inequalities in number-line order (using < or ≤ symbols). This gives a visual indication of where the variable stands in relation to the limit(s). Perhaps a more conventional way to write the domain for the second piece is, x > 5.

User Yogini
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