1) f(x) = 2x - 5
Determine the difference quotient

Difference quotient: = 2

Step-by-step explanation:

The difference quotient is the word that we use when trying to calculate the average rate of change of the function over a specific interval, which in this case, is h.

The formula for calculating the difference quotient is:

In order to determine the difference quotient of the function we need to first evaluate of this function.

We can do so by substituting x + h for x into the function.

Distribute 2 inside the parentheses.

Now that we have found f(x + h), we can use the formula for the difference quotient and substitute f(x + h) and f(x) into it:

Distribute the negative sign inside the parentheses.

Combine like terms in the numerator. The 2x and 5's cancel out. We are left with:

The h's cancel out, and this simplifies to:

The difference quotient of f(x) is 2.

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.