Answer to Question #177360 in Calculus for salma

Question

Question 3: If the domain of 𝑓(𝑥)=3𝑎𝑥+22𝑏𝑥−1 is given by 𝐷={𝑥|𝑥∈ℝ,𝑥≠3} and the graph of this function passes through the point (−3,8). Write down the values of 𝑎 and 𝑏.

Answer 1

For a given function

$$f(x)=\frac{3ax+2}{2bx-1}\to\,\,\,D=\left\{x\left|2bx-1\neq0\right.\right\}\\[0.3cm] D=\left\{x\left|x\neq\frac{1}{2b}\right.\right\}\,\,\,\text{and}\,\,\,D=\left\{x\left|x\in\mathbb{R},x\neq3\right.\right\}\left(\text{by condition}\right)\\[0.3cm] \frac{1}{2b}=3\to\boxed{b=\frac{1}{6}}$$

By the condition of the problem, the function $f(x)$ passes through the point $(-3,8)$, which means that

$$f(-3)=8\to\frac{2a\cdot(-3)+2}{2\cdot\displaystyle\frac{1}{6}\cdot(-3)-1}=8\to\\[0.3cm] \frac{2\cdot(-3a+1)}{-2}=8\to3a-1=8\to a=\frac{9}{3}=3\to\boxed{a=3}$$

Conclusion,

$$\boxed{a=3\,\,\,\text{and}\,\,\,b=\frac{1}{6}}$$

ANSWER

$$a=3\,\,\,\text{and}\,\,\,b=\frac{1}{6}$$

Q.E.D.