Question

Consider the piecewise-defined function given by the graph. What are these values?

Answer 1

Answers:

f(-3) = -1

f(-1) = 2

f(3) = 5

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Explanations:

Draw a vertical line through -3 on the x axis. Make sure this vertical line crosses the orange graph. Mark this point of intersection and then draw a horizontal line from this point to the y axis. You should find that it touches -1 on the y axis. Check out the attached image to see this process in action.

Those steps basically say that x = -3 leads to y = -1. Since y = f(x), this means f(-3) = -1

Through similar steps, you should find that f(-1) = 2 and f(3) = 5

Note that the open hole/circle does not count. So if you land on an open hole, just keep going until you hit a closed circle on the other piece of the graph.

Answer 2

A) f (–3) = -1

B) f (–1) = 2

C) f (3) = 5

Further explanation

Straight-line equations are mathematical equations that are described in the plane of cartesian coordinates

General formula

y-y1 = m (x-x1)

or

y = mx + c

Where

m = straight-line gradient which is the slope of the line

x1, y1 = the Cartesian coordinate that is crossed by the line

c = constant

The formula for a gradient (m) between 2 points in a line

m = Δy / Δx

or we can use:

$\rm \dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}$

Piecewise Functions: functions that have different equations based on the input (x) value

We look for 3 pieces of function from the graph

1. graph 1 is a straight line that passes through 2 points : (-4.0) and (-1, -3) so that the straight line equation:

$\rm \dfrac{y-0}{-3-0}=\dfrac{x+4}{-1+4}\\\\\dfrac{y}{-3}=\dfrac{x+4}{3}\\\\3y=-3(x+4)\\\\3y=-3x-12\\\\y=-x-4\Rightarrow \:for\:x$

2. graph 2 is a straight line with the equation: \

$\rm y=2\Rightarrow \:for\:-1\leq x$

3. graph 1 is a straight line that passes through 2 points,  (3.5) and (5.1) so that the straight line equation  :

$\rm \dfrac{y-5}{1-5}=\dfrac{x-3}{5-3}\\\\\dfrac{y-5}{-4}=\dfrac{x-3}{2}\\\\2(y-5)=-4(x-3)\\\\2y-10=-4x+12\\\\2y=-4x+22\\\\y=-2x+11\:\Rightarrow for\:x\geq 3$

A function made up of 3 pieces:

a solid dot means "including",

an open dot means "not including"

$\rm y=-x-4\:for\:x$

$\rm y=2\:for\:-1\leq x$

$\rm y=-2x+11\:for\:x\geq 3$

A) f (–3) =

Because -3 <-1, the function used is y = -x-4, so

f (-3) = - (- 3) -4

f (-3) = 3-4

f (-3) = -1

B) f (–1) =

Because x = -1, the function used y = 2, so

f (-1) = 2

C) f (3) =

Because x = 3, the function used is = -2x + 11, so

f (3) = -2 (3) +11

f (3) = -6 + 11

f (3) = 5

Learn more

Piecewise Functions

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the inverse of the function

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domain of the function

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