Write the equation of the piecewise function ƒ that is represented by its graph.

Piecewise function with 3 pieces: [tex]x^2+1, 4x-8, 2[/tex]

The piecewise function ƒ that is represented by the graph in the image is:

ƒ(x) =

[tex]x^2+1, & \text{if } 0 \le x < 4 \\[/tex]

[tex]4x-8, & \text{if } 4 \le x < 5 \\[/tex]

[tex]5^2+1 = 26, & \text{if } x = 5[/tex]

This is because the graph of the function consists of three distinct pieces:

For 0≤x<4, the graph is a parabola with vertex at (0,1) and opening upwards. This suggests that the function is of the form [tex]ax^{2} +bx+c.[/tex]

We can find the values of a, b, and c by substituting the points (0,1), (1,1), and (4,16) into the equation.

This gives us the system of equations:

\begin{cases}

[tex]a \cdot 0^2 + b \cdot 0 + c = 1[/tex]

[tex]a \cdot 1^2 + b \cdot 1 + c = 1 \\[/tex]

[tex]a \cdot 4^2 + b \cdot 4 + c = 16[/tex]

\end{cases}

Solving this system gives us a=1, b=0, and c=1, so the equation of the function for this interval is [tex]x^{2} +1[/tex].  

For 4≤x<5, the graph is a line with slope 4 and y-intercept −8.

This suggests that the function is of the form mx+b.

We can find the values of m and b by substituting the points (4,2) and (5,25) into the equation.

This gives us the system of equations:

\begin{cases}

[tex]4m+b = 2 \\[/tex]

5m+b = 25

\end{cases}

Solving this system gives us m=4 and b=−8, so the equation of the function for this interval is 4x−8.

For x=5, the graph is a horizontal line at y=26.

This suggests that the function is of the form c.

We can find the value of c by simply looking at the graph.

This gives us c=26, so the equation of the function for this interval is 26.

Therefore, the complete piecewise function is:

ƒ(x) =

\begin{cases}

[tex]x^2+1, & \text{if } 0 \le x < 4 \\[/tex]

[tex]4x-8, & \text{if } 4 \le x < 5 \\[/tex]

[tex]5^2+1 = 26, & \text{if } x = 5[/tex]

\end{cases}

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