Question

Can someone help me with 19-24... I need it to be worked out and answered... Please and thank you I really don't understand it, But it's due tomorrow morning so can someone please help... Thanks

Answer 1

Let me help you to understand this problem. Linear functions are those whose graph is a straight line. The form of a linear function is given by the following form:

`y=f(x)=ax+b \\ \\ where \ a \ and \ b \ are \ constants \\ \\ \\ a:Represents \ the \ slope \ of \ the \ line \\ \\ b:Represents \ the \ y-intercept`

The y-intercept occurs when x = 0.

So, our problem is to find the constants
`a \ and \ b` in each case. Therefore:

FIRST. (19)

`f(0)=2, \ f(2)=4`

The slope of a line can be found as follows:

`a=(f(x_(2))-f(x_(1)))/(x_(2)-x_(1)) \\ \\ \\ f(x_(2))=f(2)=4 \\ \\ f(x_(1))=f(0)=2 \\ \\ \\ \therefore a=(4-2)/(2-0)=1`

As the y-intercept occurs when x = 0, therefore:

`b=f(0)=2`

Accordingly, the first linear function is:

`\boxed{y=f(x)=x+2}`

SECOND. (20)

`f(0)=7, \ f(3)=1`

Applying the same reasoning as in the first exercise, the slope is:

`f(x_(2))=f(3)=1 \\ \\ f(x_(1))=f(0)=7 \\ \\ \\ \therefore a=(1-7)/(3-0)=-2`

And the y-intercept is:

`b=f(0)=7`

Accordingly, the second linear function is:

`\boxed{y=f(x)=-2x+7}`

THIRD. (21)

`f(4)=-3, \ f(0)=-2`

The slope is:

`f(x_(2))=f(4)=-3 \\ \\ f(x_(1))=f(0)=-2 \\ \\ \\ \therefore a=(-3-(-2))/(4-0)=-(1)/(4)`

And the y-intercept is:

`b=f(0)=-2`

Accordingly, the third linear function is:

`\boxed{y=f(x)=-(1)/(4)x-2}`

FOURTH. (22)

`f(5)=-1, \ f(0)=-5`

The slope is:

`f(x_(2))=f(5)=-1 \\ \\ f(x_(1))=f(0)=-5 \\ \\ \\ \therefore a=(-1-(-5))/(5-0)=(4)/(5)`

And the y-intercept is:

`b=f(0)=-5`

Accordingly, the fourth linear function is:

`\boxed{y=f(x)=(4)/(5)x-5}`

FIFTH. (23)

`f(-2)=6, \ f(0)=-4`

The slope is:

`f(x_(2))=f(-2)=6 \\ \\ f(x_(1))=f(0)=-4 \\ \\ \\ \therefore a=(6-(-4))/(-2-0)=-5`

And the y-intercept is:

`b=f(0)=-4`

Accordingly, the fifth linear function is:

`\boxed{y=f(x)=-5x-4}`

SIXTH. (24)

`f(0)=3, \ f(-6)=3`

The slope is:

`f(x_(2))=f(-6)=3 \\ \\ f(x_(1))=f(0)=3 \\ \\ \\ \therefore a=(3-3)/(-6-0)=0`

And the y-intercept is:

`b=f(0)=3`

Accordingly, the sixth linear function is:

`\boxed{y=f(x)=3} \\ \\ This \ is \ a \ constant \ function`