Questions: Using f(x)=√(x+1) and g(x)=14/(x-3) a) f(g(x)) b) g(f(x)) c) f(g(10)) d) g(f(8))

Transcript text: 2. Using $f(x)=\sqrt{x+1}$ and $g(x)=\frac{14}{x-3}$ a) $f(g(x))$ b) $g(f(x))$ c) $f(g(10))$ d) $g(f(8))$

Solution

Solution Steps

To solve the given problems, we need to find the compositions of the functions $ f(x) = \sqrt{x+1} $ and $ g(x) = \frac{14}{x-3} $.

a) For $ f(g(x)) $, substitute $ g(x) $ into $ f(x) $.

b) For $ g(f(x)) $, substitute $ f(x) $ into $ g(x) $.

c) For $ f(g(10)) $, first calculate $ g(10) $ and then substitute the result into $ f(x) $.

Solution Approach

Step 1: Calculate $ f(g(x)) $

To find $ f(g(x)) $, we first compute $ g(x) = \frac{14}{x - 3} $. Then, we substitute this into $ f(x) = \sqrt{x + 1} $: \[ f(g(x)) = f\left(\frac{14}{x - 3}\right) = \sqrt{\frac{14}{x - 3} + 1} \] For $ x = 10 $: \[ f(g(10)) = f\left(\frac{14}{10 - 3}\right) = f\left(\frac{14}{7}\right) = f(2) = \sqrt{2 + 1} = \sqrt{3} \approx 1.7321 \]

Step 2: Calculate $ g(f(x)) $

Next, we compute $ g(f(x)) $ by substituting $ f(x) $ into $ g(x) $: \[ g(f(x)) = g\left(\sqrt{x + 1}\right) = \frac{14}{\sqrt{x + 1} - 3} \] For $ x = 10 $: \[ g(f(10)) = g\left(\sqrt{10 + 1}\right) = g\left(\sqrt{11}\right) = \frac{14}{\sqrt{11} - 3} \approx 44.2164 \]

Step 3: Calculate $ f(g(10)) $

We already computed $ g(10) $: \[ g(10) = \frac{14}{10 - 3} = \frac{14}{7} = 2 \] Now substituting this into $ f(x) $: \[ f(g(10)) = f(2) = \sqrt{2 + 1} = \sqrt{3} \approx 1.7321 \]