The function of f and g are such that
f(x) = 5x + 3
g(x) = ax + b
g(3) = 20 and f^-1(33) = g(1)
Find the value of a and b
+26367 The function of f and g are such that
f(x) = 5x + 3
g(x) = ax + b
g(3) = 20 and f^-1(33) = g(1)
Find the value of a and b
1.
\(\begin{array}{|lrcll|} \hline & g(x) &=& ax+b \\\\ & g(3) &=& a\cdot 3+b \quad & | \quad g(3) = 20 \\ & 20 &=& a\cdot 3+b \\ \mathbf{(1)} & \mathbf{3a+b} & \mathbf{=} & \mathbf{20} \\ \hline \end{array}\)
2.
\(\begin{array}{|lrcll|} \hline & f(x) &=& 5x+3 \\ & y &=& 5x + 3 \\ & 5x &=& y-3 \\ & x &=& \frac{y-3}{5} \quad & | \quad \text{change $x$ and $y$ } \\ & y &=& \frac{x-3}{5} \\ & f^{-1}(x) &=& \frac{x-3}{5} \\\\ & f^{-1}(33) &=& \frac{33-3}{5} \\ & &=& \frac{30}{5} \\ & f^{-1}(33) &=& 6 \quad & | \quad f^{-1}(33) = g(1) \\ & g(1) &=& 6 \quad & | \quad g(1) = a\cdot 1 + b \\ & a\cdot 1 + b &=& 6 \\ \mathbf{(2)} & \mathbf{a+b} & \mathbf{=} & \mathbf{6} \\ \hline \end{array} \)
\(\begin{array}{|lrcll|} \hline \mathbf{(1)} & \mathbf{3a+b} & \mathbf{=} & \mathbf{20} \\ \mathbf{(2)} & \mathbf{a+b} & \mathbf{=} & \mathbf{6} \\ \\ \hline (1)-(2) & 3a+b -a -b &=& 20-6 \\ & 2a &=& 14 \quad & | \quad : 2 \\ & \mathbf{a} &\mathbf{=}& \mathbf{7} \\\\ & a+b &=& 6 \quad & | \quad a=7 \\ & 7+b &=& 6 \\ & b &=& 6-7 \\ & \mathbf{b} &\mathbf{=}& \mathbf{-1} \\ \hline \end{array}\)
a = 7 and b = -1
