document.write( "Question 895964: I believe this the right category. I was working on my calculus homework online and this question really stumped me.
\n" ); document.write( "Find
\n" ); document.write( "f(a), f(a + h),
\n" ); document.write( " and the difference quotient
\n" ); document.write( "f(a + h) − f(a)/h\r
\n" ); document.write( "\n" ); document.write( ",
\n" ); document.write( " where
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\n" ); document.write( "f(x)=4/x+9 \r
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\n" ); document.write( "\n" ); document.write( "f(a)= ? \r
\n" ); document.write( "\n" ); document.write( "f(a + h)= ?
\n" ); document.write( "f(a + h) − f(a)/h=?\r
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Algebra.Com's Answer #543122 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm going to assume the function is $\"f%28x%29+=+4%2F%28x%2B9%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "First we need f(a)\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29+=+4%2F%28x%2B9%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28a%29+=+4%2F%28a%2B9%29\"$ Replace every 'x' with 'a'\r
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\n" ); document.write( "\n" ); document.write( "Now we need f(a+h)\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29+=+4%2F%28x%2B9%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+=+4%2F%28a%2Bh%2B9%29\"$ Replace every 'x' with \"a+h\"\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+=+4%2F%28a%2B9%2Bh%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+=+4%2F%28%28a%2B9%29%2Bh%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "-------------------------------------------------------\r
\n" ); document.write( "\n" ); document.write( "Now because both f(a) and f(a+h) have an \"a+9\" in the denominator, I'm going to let z = a+9\r
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\n" ); document.write( "\n" ); document.write( "So $\"f%28a%29+=+4%2F%28a%2B9%29\"$ turns into $\"f%28a%29+=+4%2Fz\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+=+4%2F%28%28a%2B9%29%2Bh%29\"$ turns into $\"f%28a%2Bh%29+=+4%2F%28z%2Bh%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "this will make the work shown below a bit more easier\r
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\n" ); document.write( "\n" ); document.write( "Subtract the two. We do so by finding the LCD (in this case z(z+h)) and then getting each fraction to have this LCD\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+4%2F%28z%2Bh%29+-+4%2Fz\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+%284z%29%2F%28z%28z%2Bh%29%29+-+4%2Fz\"$ Multiply top and bottom of the first fraction by 'z' \r
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Multiply top and bottom of the second fraction by 'z+h' \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+%284z%29%2F%28z%28z%2Bh%29%29+-+%284z%2B4h%29%2F%28z%28z%2Bh%29%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+%284z+-+%284z%2B4h%29%29%2F%28z%28z%2Bh%29%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+%284z+-+4z-4h%29%2F%28z%28z%2Bh%29%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+-%284h%29%2F%28z%28z%2Bh%29%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+-%284h%29%2F%28%28a%2B9%29%28a%2B9%2Bh%29%29\"$ Replace every 'z' with 'a+9'\r
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\n" ); document.write( "\n" ); document.write( "--------------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "The last step is to divide all of that by 'h'. Put another way, we multiply all of that by $\"1%2Fh\"$ . This is because dividing is the same as multiplying by the reciprocal.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28a%2Bh%29+-+f%28a%29+=+-%284h%29%2F%28%28a%2B9%29%28a%2B9%2Bh%29%29\"$ \r
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Multiply both sides by $\"1%2Fh\"$ \r
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We have these common 'h' terms (in top and bottom)\r
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which divide to 1 and cancel out (they go away)\r
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\n" ); document.write( "\n" ); document.write( " $\"%28f%28a%2Bh%29+-+f%28a%29%29%2Fh+=+-4%2F%28%28a%2B9%29%28a%2B9%2Bh%29%29%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "which is the final answer. You could expand out the denominator, but that makes things messier than they have to be. \n" ); document.write( "

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