Questions: For the following function find (a) f(9), (b) f(-1/2), (c) f(a), (d) f(2/m), and (e) any values of x such that f(x)=1. f(x) = (2x+6)/(x-9) if x ≠ 9 7 if x=9 (c) Find the value of f(a). f(a) = (2a+6)/(a-9) if a ≠ 9 7 if a=9 (d) Find the value of f(2/m). f(2/m) = (4+6m)/(2-9m) if m ≠ 2/9 7 if m=2/9 (e) Find the values of x such that f(x)=1. x =

For the following function find (a) f(9), (b) f(-1/2), (c) f(a), (d) f(2/m), and (e) any values of x such that f(x)=1.
f(x) = (2x+6)/(x-9) if x ≠ 9
7 if x=9
(c) Find the value of f(a).
f(a) = (2a+6)/(a-9) if a ≠ 9
7 if a=9
(d) Find the value of f(2/m).
f(2/m) = (4+6m)/(2-9m) if m ≠ 2/9
7 if m=2/9
(e) Find the values of x such that f(x)=1.
x =

Transcript text: For the following function find (a) $f(9)$, (b) $f\left(-\frac{1}{2}\right)$, (c) $f(a)$, (d) $f\left(\frac{2}{m}\right)$, and (e) any values of $x$ such that $f(x)=1$. \[ f(x)=\left\{\begin{array}{ll} \frac{2 x+6}{x-9} & \text { if } x \neq 9 \\ 7 & \text { if } x=9 \end{array}\right. \] (c) Find the value of $\mathrm{f}(\mathrm{a})$. \[ f(a)=\left\{\begin{array}{ll} \frac{2 a+6}{a-9} & \text { if } a \neq 9 \\ 7 & \text { if } a=9 \end{array}\right. \] (d) Find the value of $f\left(\frac{2}{m}\right)$. \[ f\left(\frac{2}{m}\right)=\left\{\begin{array}{ll} \frac{4+6 m}{2-9 m} & \text { if } m \neq \frac{2}{9} \\ 7 & \text { if } m=\frac{2}{9} \end{array}\right. \] (e) Find the values of x such that $\mathrm{f}(\mathrm{x})=1$. \[ \mathrm{x}=\square \]

Solution

Solution Steps

Step 1: Evaluate $ f(9) $

Since $ x = 9 $, we use the condition for $ f(x) $ when $ x = 9 $: \[ f(9) = 7 \]

Step 2: Evaluate $ f\left(-\frac{1}{2}\right) $

For $ x = -\frac{1}{2} $, which is not equal to 9, we use the other condition: \[ f\left(-\frac{1}{2}\right) = \frac{2\left(-\frac{1}{2}\right) + 6}{-\frac{1}{2} - 9} = \frac{-1 + 6}{-\frac{1}{2} - 9} = \frac{5}{-\frac{19}{2}} = -\frac{10}{19} \]

Step 3: Evaluate $ f(a) $

For $ f(a) $, we consider the case when $ a \neq 9 $: \[ f(a) = \frac{2a + 6}{a - 9} \]

Step 4: Evaluate $ f\left(\frac{2}{m}\right) $

For $ m \neq \frac{2}{9} $, we have: \[ f\left(\frac{2}{m}\right) = \frac{4 + 6m}{2 - 9m} \]

Step 5: Solve for $ x $ such that $ f(x) = 1 $

To find $ x $ such that $ f(x) = 1 $, we set up the equation: \[ \frac{2x + 6}{x - 9} = 1 \] Cross-multiplying gives: \[ 2x + 6 = x - 9 \] Rearranging leads to: \[ 2x - x = -9 - 6 \implies x = -15 \]

Final Answer

(a) $ \boxed{7} $
(b) $ \boxed{-\frac{10}{19}} $
(c) $ f(a) = \frac{2a + 6}{a - 9} $
(d) $ f\left(\frac{2}{m}\right) = \frac{4 + 6m}{2 - 9m} $
(e) $ \boxed{-15} $

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