In order to determine the relation between $f (22)$ and $g (22),$ we must find the equation of each function by using the information given in the tables.

Finding the Equation of $f (x)$

The table below gives us some points on the graph of $f (x) .$

$x$	$f (x)$
$- 5$	$- 23$
$- 4$	$- 20$
$- 3$	$- 17$
$- 2$	$- 14$

Next, we plot all these points on a coordinate plane.

Now, let's find the equation of the line that passes through

(- 5, - 23)

and

(- 3, - 17) .

First, we will find its slope.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

We substitute

(x_{1}, y_{1}) = (- 5, - 23)

and

(x_{2}, y_{2}) = (- 3, - 17)

into the formula above.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstituteValues

Substitute values

m = \frac{- 1 7 - ( - 2 3 )}{- 3 - ( - 5 )}

▼

Simplify right-hand side

NegNeg

$- (- a) = a$

m = \frac{- 1 7 + 2 3}{- 3 + 5}

AddTerms

Add terms

m = \frac{6}{2}

CalcQuot

Calculate quotient

m = 3

We can now use the point-slope form

y - y_{1} = m (x - x_{1})

to find the equation of the line.

y - y_{1} = m (x - x_{1})

SubstituteIII

Substitute $(x_{1}, y_{1}) = (- 5, - 23) & m = 3$

y - (- 23) = 3 (x - (- 5))

▼

Solve for

y

NegNeg

$- (- a) = a$

y + 23 = 3 (x + 5)

Distr

Distribute $3$

y + 23 = 3 x + 15

SubEqn

$LHS - 23 = RHS - 23$

y = 3 x - 8

Therefore, the equation of the function

f (x)

is

f (x) = 3 x - 8 .

We are ready to find

f (22) .

f (22) = 3 (22) - 8 = 58

Finding the Equation of $g (x)$

Now, let's take a look at the right-hand side table, corresponding to $g (x) .$

$x$	$g (x)$
$- 2$	$- 18$
$- 1$	$- 14$
$0$	$- 10$
$1$	$- 6$

As before, let's plot these points on a coordinate plane.

This time we already have the

y -

intercept, namely

b = - 10 .

To find the equation, we will use slope-intercept form.

y = m x - 10

Next, we will use points

(0, - 10)

and

(1, - 6)

to find the slope.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(0, - 10)$ & $(1, - 6)$

m = \frac{- 6 - ( - 1 0 )}{1 - 0}

▼

Simplify right-hand side

NegNeg

$- (- a) = a$

m = \frac{- 6 + 1 0}{1 - 0}

AddSubTerms

Add and subtract terms

m = \frac{4}{1}

CalcQuot

Calculate quotient

m = 4

Consequently, the equation of the function

g (x)

is

g (x) = 4 x - 10 .

We can now find the value of

g (22) .

g (22) = 4 (22) - 10 = 80

After finding both values, we can state the relation between them.

f (22) 58 < < g (22) 80

Finding the Equation of f(x)

Finding the Equation of g(x)

Finding the Equation of $f (x)$

Finding the Equation of $g (x)$

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