[tex]14)f:\mathbb{R}\rightarrow\mathbb{R} \: \: \: ,f(x)= {x}^{2} + 2x - 5[/tex]
[tex]A(1,m)\:\in\:G_{f} = > f(1) = m[/tex]
[tex]m = {1}^{2} + 2 \times 1 - 5 = - 2[/tex]
[tex]15)f:\mathbb{R}\rightarrow\mathbb{R} \: \: \: ,f(x) = {x}^{2} + 5x - 5[/tex]
[tex]A(m,1)\:\in\:G_{f} = > f(m) = 1[/tex]
[tex] {m}^{2} + 5m - 5 = 1[/tex]
[tex] {m}^{2} + 5m - 5 - 1 = 0[/tex]
[tex] {m}^{2} + 5m - 6 = 0[/tex]
[tex] {m}^{2} + 6m - m - 6 = 0[/tex]
[tex]m(m + 6) - (m + 6) = 0[/tex]
[tex](m + 6)(m - 1) = 0[/tex]
[tex]m + 6 = 0 = > m_{1} = - 6[/tex]
[tex]m - 1 = 0 = > m_{2} = 1[/tex]
[tex]16)f,g:\mathbb{R}\rightarrow\mathbb{R}[/tex]
[tex]a)f(x) = 2x - 5[/tex]
[tex]g(x) = x + 3[/tex]
[tex]2x - 5 = x + 3[/tex]
[tex]2x - x = 3 + 5[/tex]
[tex]x = 8[/tex]
[tex]f(8) = 2 \times 8 - 5 = 16 - 5 = 11[/tex]
[tex] = > A(8,11)[/tex]
[tex]b)f(x) = x - 1[/tex]
[tex]g(x) = {x}^{2} - 5x + 7[/tex]
[tex]x - 1 = {x}^{2} - 5x + 7[/tex]
[tex] {x}^{2} - 5x - x + 7 + 1 = 0[/tex]
[tex] {x}^{2} - 6x + 8 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 6[/tex]
[tex]c = 8[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 6)}^{2} - 4 \times 1 \times 8[/tex]
[tex]\Delta = 36 - 32[/tex]
[tex]\Delta = 4[/tex]
[tex]x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a} = \frac{ - ( - 6) \pm \sqrt{4} }{2 \times 1} = \frac{6 \pm2}{2} [/tex]
[tex]x_{1} = \frac{6 + 2}{2} = \frac{8}{2} = 4[/tex]
[tex] = > f(x_{1}) = 4 - 1 = 3 = > B(4,3)[/tex]
[tex]x_{2} = \frac{6 - 2}{2} = \frac{4}{2} = 2[/tex]
[tex]=>f(x_{2}) = 4 - 2 = 2 = > C(2,2)[/tex]
