How do you find the domain and range and intercepTs for #R(x) = (x^2 + x - 12)/(x^2 - 4)#?

Question

Savannah Anderson · Accepted Answer

see explanation.

To find any #color(blue)"excluded values"# on the domain. The denominator of R(x) cannot be zero as this would make R(x) undefined. Evaluating the denominator to zero and solving gives the values that x cannot be. #"solve "x^2-4=0rArrx^2=4rArrx=+-2# #rArr"domain is " x inRR,x!=+-2# To find #color(blue)"excluded values"# on the range. divide all terms on numerator/denominator by the highest power of x, that is #x^2# #R(x)=(x^2/x^2+x/x^2-(12)/x^2)/(x^2/x^2-4/x^2)=(1+1/x-(12)/x^2)/(1-4/x^2)# as #xto+-oo,1/x,(12)/x^2,4/x^2to0# #lim_(xto+-oo),R(x)to(1+0-0)/(1-0)# #=1/1=1larrcolor(red)" excluded value"# #"range is " y inRR,y!=1# #color(blue)"intercepts"# #• " let x = 0, in equation, for y-intercept"# #• " let y = 0, in equation, for x-intercepts"# #x=0toy=(-12)/(-4)=3larrcolor(red)" y-intercept"# The numerator is the only part of R ( x ) that can equal zero when y = 0 #rArry=0tox^2+x-12=0# #rArr(x+4)(x-3)=0# #rArrx=-4,x=3larrcolor(red)" x-intercepts"# graph{(x^2+x-12)/(x^2-4) [-10, 10, -5, 5]}

Eli Assouline · Answer

undefined To find the domain of R(x), we need to determine the values of x for which the function is defined. In this case, the function R(x) is defined for all real numbers except the values that make the denominator equal to zero. Thus, the domain of R(x) is all real numbers except x = 2 and x = -2.

To find the range of R(x), we need to determine the set of all possible output values. Since the function is a rational function, the range will be all real numbers except for any vertical asymptotes or holes in the graph. In this case, there are no vertical asymptotes or holes, so the range of R(x) is all real numbers.

To find the x-intercepts, we set R(x) equal to zero and solve for x. In this case, we have (x^2 + x - 12)/(x^2 - 4) = 0. By factoring the numerator and denominator, we get (x + 4)(x - 3)/(x + 2)(x - 2) = 0. Setting each factor equal to zero, we find x = -4 and x = 3 as the x-intercepts.

To find the y-intercept, we set x = 0 in the function R(x). Plugging in x = 0, we get R(0) = (0^2 + 0 - 12)/(0^2 - 4) = -12/-4 = 3. Therefore, the y-intercept is (0, 3).