in this lesson we're going to focus on evaluating piecewise functions a piecewise function is a function that can be broken up into many parts so this particular piecewise function can be equal to 4x plus 5 or 3x minus 8 depending on the x value so let's say it's equal to 4x plus 5 when x is less than 2 and it's equal to 3x minus 8 when x is equal to or greater than two so go ahead and find the value of f of negative two f of two and f of five feel free to pause the video and work on this problem so let's evaluate the function when x is negative 2. so should we use this portion of the piecewise function or the bottom part should we use 4x plus 5 or 3x minus 8. negative two is less than positive two it's not equal to or greater than positive two so therefore negative two corresponds to this range so we need to use the first part of the piecewise function so let's replace x with negative two four times negative two is negative eight negative eight plus five is negative three so f of negative two is negative three now what about f of positive two should we use four x plus five or three x minus eight now x is equal to two in this equality not in this one so we have to use three x minus eight so it's going to be three times two minus eight three times two is six six minus eight is negative two and that's the answer now what is the function value at five so once again five is greater than two so we need to use three x minus eight so it's gonna be three times five minus eight three times five is fifteen fifteen minus 8 is 7. and that's it let's work on some more examples so let's say if we have the function f of x and it's equal to x squared plus three x minus seven when x is less than negative one and it's equal to five x plus six when x is greater than or equal to negative one actually let me change that let's say when x is greater than or equal to negative one but less than two and let's say it's uh equal to x cubed plus four when x is greater than two and is equal to twelve when x equals two so with this information i want you to evaluate f of negative 4 f of let's say 0 f of two and f of three so feel free to pause the video and try that so let's evaluate the function at x equals negative four so negative four is less than negative one therefore we need to use x squared plus three x minus seven so this is going to be negative four squared plus three times negative four minus seven negative four squared or negative four times negative four that's sixteen three times negative four is negative twelve and six minus twelve is four four minus seven is negative three so f of negative four is equal to negative three so that's the first answer i believe my math is correct i don't think i made any mistakes on that now let's evaluate f of zero zero is between negative one and two so we need to use five x plus six so this is going to be five times zero plus six five times zero is zero zero plus six is six now what about the next one f of 2. now when x is exactly 2 the function is equal to 12. so f of 2 is 12. there's no math involved in that step now what about the last one f of three when x is three we need to use x cubed plus four because that's when x is greater than two when x is three so therefore this is going to be 3 raised to the third power plus 4. so 3 to the third power is 27 27 plus 4 is equal to 31 and that covers that problem you