Write an equation for the linear function f with the given values.

you are given 2 points:

(1,3) and (3,4)

find the slope,
m = (4-3)/(3-1) = 1/2

then
y-3 = (1/2)(x - 3)

arrange it into one of the standard ways to write a straight-line
equation.

y=1/2x+3/2

Sure, let's put the clown nose on and get to work!

To find the equation for the linear function f, we need to determine the slope (m) and the y-intercept (b). Once we have those values, we can use the equation y = mx + b.

Given that f(1) = 3 and f(3) = 4, we can use these two points to find the slope:

m = (y2 - y1) / (x2 - x1)
= (4 - 3) / (3 - 1)
= 1/2

Now, we need to find the y-intercept, which is the value of f(0). We can use the slope-intercept form of a line equation to solve for b:

f(x) = mx + b
3 = (1/2)(1) + b
3 = 1/2 + b
b = 3 - 1/2
b = 5/2

So, the equation for the linear function f is:

f(x) = (1/2)x + 5/2

Just remember, if you're having trouble with the equation, I can always juggle some numbers for you!

To write an equation for the linear function f, we need to find the slope and the y-intercept.

First, let's find the slope. The formula for slope is:
slope = (change in y) / (change in x)

We can use the values of f(1) and f(3) to find the change in y and change in x.
change in y = f(3) - f(1) = 4 - 3 = 1
change in x = 3 - 1 = 2

Therefore, the slope is 1/2.

Next, we need to find the y-intercept. We can use the formula:
y = mx + b

Since we know the slope is 1/2 and we have a point (1, 3), we can substitute these values to find the y-intercept:
3 = (1/2)(1) + b
3 = 1/2 + b

To isolate b, subtract 1/2 from both sides:
3 - 1/2 = b
6/2 - 1/2 = b
5/2 = b

So, the y-intercept is 5/2.

Putting it all together, the equation for the linear function f is:
f(x) = (1/2)x + 5/2

To find an equation for the linear function f, we can use the point-slope form of a linear equation.

The point-slope form is given by:
y - y₁ = m(x - x₁),

where (x₁, y₁) are the coordinates of a point on the line, and m is the slope of the line.

Given that f(1) = 3, we have the point (1, 3).

To find the value of m (the slope), we can use the second point given, f(3) = 4, which corresponds to the coordinates (3, 4).

Using these two points, we can substitute the values of (x₁, y₁) = (1, 3) and (x, y) = (3, 4) into the point-slope form equation:

y - 3 = m(x - 1).

Substitute the y value as 4 and the x value as 3:

4 - 3 = m(3 - 1).

1 = 2m.

Now, solve for m:

m = 1/2.

We have the slope m = 1/2, and we can use the coordinates (1, 3) to find the equation:

y - 3 = (1/2)(x - 1).

Simplify the equation:

y - 3 = 1/2x - 1/2.

Add 3 to both sides:

y = 1/2x + 5/2.

Therefore, the equation for the linear function f with the given values is f(x) = 1/2x + 5/2.