Math 104 - Problems for the gateway exam

(1) $\mbox{Evaluate the function } f(x)=3x^2-2x+1 \mbox{ for } x=3.$
(2) $\mbox{Evaluate the function } g(t)=2t^2-2 \mbox{ for } t=a+h.$
(3) $\mbox{Evaluate the function } \displaystyle{f(y)=\frac{6y-1}{y}} \mbox{ for } y=c+1.$


(4) $\mbox{Evaluate the function } f(x)=x^2-6x \mbox{ for } x=b-1.$
(5) $\mbox{Evaluate the function } v(t)=3t+2 \mbox{ for } t=a+h.$
(6) $\mbox{Evaluate the function } \displaystyle{h(s)=3-s-\frac{1}{2}s^2} \mbox{ for } s=j-2.$


(7) $\mbox{Evaluate the function } f(x)=2x^2-5 \mbox{ for } x=-1.$
(8) $\mbox{Evaluate the function } g(t)=8t-3 \mbox{ for } t=d-2.$
(9) $\mbox{Evaluate the function } \displaystyle{f(y)=\frac{y+2}{y-2}} \mbox{ for } y=m+k.$


(10) $\mbox{Evaluate the function } f(x)=9x+1 \mbox{ for } x=t+2.$



(11) $\mbox{Solve }P=S-Srt \mbox{ for } r$.
(12) $\mbox{Solve }2rx+7=8(r-x) \mbox{ for } x$.
(13) $\mbox{Solve }\displaystyle{\frac{1}{f}=\frac{1 }{d_0}+\frac{1}{d_1}} \mbox{ for } f$.


(14) $\mbox{Solve }2ax-3d=b(x-a) \mbox{ for } x$.
(15) $\mbox{Solve }\displaystyle{v=\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}} \mbox{ for } v_1$.


(16) $\mbox{Solve }x+y=\sqrt{x^2+y^2+1} \mbox{ for } y$.
(17) $\mbox{Solve }\displaystyle{\frac{1}{x}+\frac{1}{y}=1} \mbox{ for } y$.


(18) $\mbox{Solve }\displaystyle{Q_w=m_wc_w(T_f-T_w)} \mbox{ for } T_w$.
(19) $\mbox{Solve }\displaystyle{y-y_1=m(x-x_1)} \mbox{ for } x$.
(20) $\mbox{Solve }\displaystyle{\frac{x}{a}+\frac{y}{b}=1} \mbox{ for } x$.



(21) $\mbox{Solve }\displaystyle{x^\frac{1}{3}-2x^\frac{1}{6}-15=0} \mbox{ for } x$.
(22) $\mbox{Solve }(2x-5)^3-(2x-5)=0 \mbox{ for } x$.
(23) $\mbox{Solve }-14x^\frac{1}{2}=x+49 \mbox{ for } x$.
(24) $\mbox{Solve }(x+4)^3=125 \mbox{ for } x$.
(25) $\mbox{Solve }\displaystyle{t^\frac{7}{2}-4t^\frac{5}{2}=-4t^\frac{3}{2}} \mbox{ for } t$.
(26) $\mbox{Solve }(x-5)^3-27=0 \mbox{ for } x$.
(27) $\mbox{Solve }\displaystyle{(h-1)^\frac{1}{3}+4(h-1)^\frac{1}{6}+3=0} \mbox{ for } h$.
(28) $\mbox{Solve }\displaystyle{2p^\frac{1}{2}=24} \mbox{ for } p$.
(29) $\mbox{Solve }12q-21q^2-6q^3=0 \mbox{ for } q$.
(30) $\mbox{Solve } \displaystyle{8z^\frac{1}{2}+14z^\frac{3}{2}-15z^\frac{5}{2}=0} \mbox{ for } z$.



(31) $\mbox{Solve }\sqrt{x-4}-6=0 \mbox{ for } x$.
(32) $\mbox{Solve }\sqrt{6-y}+\sqrt{5y+6}=6 \mbox{ for } y$.
(33) $\mbox{Solve }b=\sqrt{12b-35} \mbox{ for } b$.
(34) $\mbox{Solve }c=3+\sqrt{3-c} \mbox{ for } c$.
(35) $\mbox{Solve }\sqrt{2x+11}-\sqrt{2x-5}=2 \mbox{ for } x$.
(36) $\mbox{Solve }\sqrt{m+7}+\sqrt{m-5}=6 \mbox{ for } m$.
(37) $\mbox{Solve }2x=\sqrt{4x+15} \mbox{ for } x$.
(38) $\mbox{Solve }\sqrt{10-t}=4 \mbox{ for } t$.
(39) $\mbox{Solve }r=\sqrt{5-r}+5 \mbox{ for } r$.
(40) $\mbox{Solve }\sqrt{x-7}+11=12 \mbox{ for } x$.



(41) $\mbox{Solve }a^4-9a^2=-14 \mbox{ for } a$.
(42) $\mbox{Solve }2x^4-11x^2+12=0 \mbox{ for } x$.
(43) $\mbox{Solve }\displaystyle{(\frac{g-1}{g})^2-10(\frac{g-1}{g})+9=0 \mbox{ for } g}$.


(44) $\mbox{Solve }6u^4-7u^2+2=0 \mbox{ for } u$.
(45) $\mbox{Solve }9x^4=30x^2-25 \mbox{ for } x$.
(46) $\mbox{Solve } \displaystyle{(\frac{f+2}{f})^2-3(\frac{f+2}{f})+2=0} \mbox{ for } f$.


(47) $\mbox{Solve }\displaystyle{6(\frac{x+4}{x})^2+5(\frac{x+4}{x})+1=0} \mbox{ for } x$.


(48) $\mbox{Solve }x^4-8x^2+12=0 \mbox{ for } x$.
(49) $\mbox{Solve }\displaystyle{4(\frac{g}{g+1})^2-4(\frac{g}{g+1})+1=0} \mbox{ for } g$.


(50) $\mbox{Solve }\displaystyle{2(\frac{x}{x-1})^2-5(\frac{x}{x-1})-3=0} \mbox{ for } x$.



(51) $\mbox{Solve }e^{3x}e^{3x}=(e^x)^x e^{-7} \mbox{ for } x$.
(52) $\mbox{Solve }2^{5t+1}=2^{t^2+7} \mbox{ for } t$.
(53) $\mbox{Solve }7^{4r+4}=7^{r^2}7^8 \mbox{ for } r$.
(54) $\mbox{Solve }\ln(3x-5)=\ln11+\ln2 \mbox{ for } x$.
(55) $\mbox{Solve }\ln(x+9)-\ln(x)=1 \mbox{ for } x$.
(56) $\mbox{Solve }(e^{2m})^{4m}=e^{3-2m} \mbox{ for } m$.
(57) $\mbox{Solve }\displaystyle{\ln(4p)+\ln(p+\frac{7}{4})=\ln2} \mbox{ for } p$.
(58) $\mbox{Solve }(5^{5x})^x=(5^{25})^x \mbox{ for } x$.
(59) $\mbox{Solve }\ln x=\ln8-2\ln x \mbox{ for } x$.
(60) $\mbox{Solve }\displaystyle{\ln(3x)+\ln(x-\frac{2}{3})=\frac{1}{2}\ln64} \mbox{ for } x$.



(61) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } (f-g)(4)$.
(62) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } (\frac{g}{f})(a)$.
(63) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } 3g(c)$.
(64) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } f(g(a+h))$.
(65) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } (f+g)(x)$.
(66) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } 2f(1)$.
(67) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } g(f(x+y))$.
(68) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } (gf)(x)$.
(69) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } g(f(\sqrt{2}))$.
(70) $\mbox{Given } f(x)=x^2+2 \mbox{ and } g(x)=\sqrt{x}-2 \mbox{, find the value of } (g-f)(9)$.



(71) $\mbox{Given } f(x)=3x-2 \mbox{ and } g(x)=x+1 \mbox{, find } f(g(x))$.
(72) $\mbox{Given } \displaystyle{g(x)=\frac{1}{x+3}} \mbox{ and } f(x)=\sqrt{x} \mbox{, find } f(g(x))$.


(73) $\mbox{Given } \displaystyle{g(x)=\frac{x-1}{x+1}} \mbox{ and } f(x)=x^2 \mbox{, find } f(g(x))$.


(74) $\mbox{Given } f(x)=x^2+4x-5 \mbox{ and } g(x)=x-c \mbox{, find } f(g(x))$.
(75) $\mbox{Given } g(x)=\sqrt{x^2-5x} \mbox{ and } f(x)=x^2+1 \mbox{, find } f(g(x))$.
(76) $\mbox{Given } \displaystyle{g(x)=\frac{3}{x}-x} \mbox{ and } \displaystyle{f(x)=\frac{x}{3}+x} \mbox{, find } f(g(x))$.


(77) $\mbox{Given } f(x)=x^\frac{1}{3}+x^\frac{1}{2} \mbox{ and } g(x)=x^3 \mbox{, find } f(g(x))$.
(78) $\mbox{Given } f(x)=5x^2-3x^\frac{1}{2} \mbox{ and } g(x)=x^4 \mbox{, find } f(g(x))$.
(79) $\mbox{Given } \displaystyle{f(x)=9x+\frac{2}{x-1}} \mbox{ and } g(x)=1+2x \mbox{, find } f(g(x))$.


(80) $\mbox{Given } g(x)=5x^2-2 \mbox{ and } f(x)= \sqrt{x}+1 \mbox{, find } f(g(x))$.



(81) $\mbox{Find the inverse function of }\displaystyle{g(t)=\frac{1}{t-1}}$.
(82) $\mbox{Find the inverse function of }f(x)=\sqrt{3x+4}$.
(83) $\mbox{Find the inverse function of }\displaystyle{v(t)=\frac{t+3}{t-2}}$.


(84) $\mbox{Find the inverse function of }\displaystyle{u(t)=\frac{4}{\sqrt{3t}}}$.


(85) $\mbox{Find the inverse function of }g(y)=\sqrt{y}+1$.
(86) $\mbox{Find the inverse function of }\displaystyle{f(x)=\frac{1}{2x+1}}$


(87) $\mbox{Find the inverse function of }\displaystyle{m(t)=\frac{4t+5}{2t}}$.


(88) $\mbox{Find the inverse function of }\displaystyle{y(x)=\frac{x^3-1}{x^3+5}}$.


(89) $\mbox{Find the inverse function of }\displaystyle{f(s)=\frac{-3}{2s+5}}$.


(90) $\mbox{Find the inverse function of }u(r)=5+\sqrt{3r-2}$.



(91) $\mbox{Simplify as much as possible }\displaystyle{\frac{(x^2+1)(x-1)^2}{x^4-1}}$.


(92) $\mbox{Simplify as much as possible }\displaystyle{\frac{2}{x+1}+\frac{2}{x-1}+\frac{1}{x^2-1}}$.

(93) $\mbox{Simplify as much as possible }\displaystyle{\sqrt{2x-1}-\frac{x+2}{\sqrt{2x-1}}}$.


(94) $\mbox{Simplify as much as possible }\displaystyle{\frac{xy+zy}{x^2+2xz+z^2}}$.


(95) $\mbox{Simplify as much as possible }\displaystyle{\frac{x^2+xy}{x^2+xy-2x-2y}}$.


(96) $\mbox{Simplify as much as possible }\displaystyle{\frac{2x^3-6x^2+x-3}{x-3}}$.

(97) $\mbox{Simplify as much as possible }\displaystyle{\frac{x^3+5x^2+6x}{x^3-4x}}$.


(98) $\mbox{Simplify as much as possible }\displaystyle{\frac{x}{x+y}-\frac{y}{x}}$.


(99) $\mbox{Simplify as much as possible }\displaystyle{\frac{x+h}{x+h+1}-\frac{x}{x+1}}$.


(100) $\mbox{Simplify as much as possible }\displaystyle{\frac{-1}{x}+\frac{2}{x^2+1}+\frac{1}{x^3+x}}$.