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Question: How do you write \(f\left( x \right)=\left| -18x-17 \right|\) as piecewise functions?...

How do you write f(x)=∣−18x−17∣f\left( x \right)=\left| -18x-17 \right| as piecewise functions?

Explanation

Solution

Before solving the above question let me tell you the definition of piecewise function. In mathematics, a piecewise-defined function is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself.

Complete step-by-step answer:
Now the given equation is:
f(x)=∣−18x−17∣f\left( x \right)=\left| -18x-17 \right|
When writing piecewise equations for absolute value functions, you will have two equations, one to the left of the vertex and the other to the right of the vertex. So, it makes sense that we find the location of the vertex.
Since we can easily see there is no vertical transformation, so the vertex lies only on the x-axis.
So the value of function must be equal to zero
⇒∣−18x−17∣=0 ⇒−18x−17=0 \begin{aligned} & \Rightarrow \left| -18x-17 \right|=0 \\\ & \Rightarrow -18x-17=0 \\\ \end{aligned}
Now subtracting the above equation by 1717 on both sides, we get
⇒−18x=17\Rightarrow -18x=17
Now divide the both sides of the above equation by−18-18
⇒x=−1718\Rightarrow x=\dfrac{-17}{18}
Now we know the x-intercept of the graph is:
⇒(−1718,0)\Rightarrow \left( \dfrac{-17}{18},0 \right)
All that is left to do is to find another point that the function passes through. It means that we have to find the y-intercept where x is equals to zero.
Now,
⇒y=∣−18(0)−17∣ ⇒y=17 \begin{aligned} & \Rightarrow y=\left| -18\left( 0 \right)-17 \right| \\\ & \Rightarrow y=17 \\\ \end{aligned}
We get (0,17)\left( 0,17 \right) the other point of the function.
Now we know that two points, we can find the equation of the line to the right of the x-intercept.
We know the formula of slope is:
⇒m=y2−y1x2−x1\Rightarrow m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
Now putting values, we get
⇒m=17−00−(−1718) ⇒m=171718 ⇒m=18 \begin{aligned} & \Rightarrow m=\dfrac{17-0}{0-\left( \dfrac{-17}{18} \right)} \\\ & \Rightarrow m=\dfrac{17}{\dfrac{17}{18}} \\\ & \Rightarrow m=18 \\\ \end{aligned}
Now by point slope form, we get the another equation of the above function,
⇒y−y1=m(x−x1)\Rightarrow y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)
Now putting the values we get,
⇒y−17=18(x−0) ⇒y−17=18x ⇒y=18x+17 \begin{aligned} & \Rightarrow y-17=18\left( x-0 \right) \\\ & \Rightarrow y-17=18x \\\ & \Rightarrow y=18x+17 \\\ \end{aligned}
Now we have our first piecewise equation:
⇒y=18x+17\Rightarrow y=18x+17, where x≥−1718x\ge \dfrac{-17}{18}
Now we will find our second piecewise equation. To find the second piecewise equation simply multiply the above equation by -1, then we get
⇒y=−18x−17\Rightarrow y=-18x-17, where x<−1718x<\dfrac{-17}{18}
Hence, by using piecewise function we get: y=18x+17y=18x+17, x≥−1718x\ge \dfrac{-17}{18} and y=−18x−17y=-18x-17, x<−1718x<\dfrac{-17}{18}

Note: Before solving these types of equations first check the given is lies on which axis, and then according to that do your solution. It is quite a lengthy process but it is simple too. To solve these types of questions must remember the slope intercept formula.