<span><span>The answer (<span>g<span>(x)</span></span>)</span>=x−7</span>
<span><span>g<span>(<span>f<span>(x)</span></span>)</span></span>=<span>√<span><span>x2</span>−7</span></span></span>
Explanation:
Put g inside f to find <span>f<span>(<span>g<span>(x)</span></span>)</span></span>
<span><span>f<span>(x)</span></span>=<span>x2</span>−4</span>
<span><span>g<span>(x)</span></span>=<span>√<span>x−3</span></span></span>
<span><span>f<span>(<span>g<span>(x)</span></span>)</span></span>=<span><span>(<span>√<span>x−3</span></span>)</span>2</span>−4</span>
<span><span>f<span>(<span>g<span>(x)</span></span>)</span></span>=x−3−4</span>
<span><span>f<span>(<span>g<span>(x)</span></span>)</span></span>=x−7</span>
Put f inside g to find <span>g<span>(<span>f<span>(x)</span></span>)</span></span>
<span><span>g<span>(x)</span></span>=<span>√<span>x−3</span></span></span>
<span><span>f<span>(x)</span></span>=<span><span>x2</span>−4</span></span>
<span><span>g<span>(<span>f<span>(x)</span></span>)</span></span>=<span>√<span><span><span>x2</span>−4</span>−3</span></span></span>
<span><span>g<span>(<span>f<span>(x)</span></span>)</span></span>=<span>√<span><span>x2</span>−7</span></span></span>:
The answers are <span><span>f<span>(<span>g<span>(x)</span></span>)</span></span>=x−7</span>
and <span><span>g<span>(<span>f<span>(x)</span></span>)</span></span>=<span>√<span><span>x2</span>−7</span></span></span>
Explanation:
This is a composition of functions.
<span><span>f<span>(x)</span></span>=<span>x2</span>−4</span>
<span><span>g<span>(x)</span></span>=<span>√<span>x−3</span></span></span>
<span>fo<span>g<span>(x)</span></span>=<span>f<span>(<span>g<span>(x)</span></span>)</span></span>=<span>f<span>(<span>√<span>x−3</span></span>)</span></span>=<span><span>(<span>√<span>x−3</span></span>)</span>2</span>−4</span>
<span>=x−3−4=x−7</span>
And
<span>go<span>f<span>(x)</span></span>=<span>g<span>(<span>f<span>(x)</span></span>)</span></span>=<span>g<span>(<span>x2</span>−4)</span></span>=<span>√<span><span>(<span>x2</span>−4)</span>−3</span></span></span>
<span>=<span>√<span><span>x2</span>−4−3</span></span>=<span>√<span><span>x2</span>−7</span></span></span>
You can see that
<span><span>f<span>(<span>g<span>(x)</span></span>)</span></span>≠</span>g(f(x))
