Questions: Find and simplify the expression if f(x)=x^2-10. f(4+h)-f(4) f(4+h)-f(4)=

Find and simplify the expression if f(x)=x^2-10. f(4+h)-f(4) f(4+h)-f(4)=
Transcript text: Find and simplify the expression if $f(x)=x^{2}-10$. \[ f(4+h)-f(4) \] \[ f(4+h)-f(4)= \]
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Solution

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Solution Steps

Step 1: Substitute \( f(4+h) \) and \( f(4) \) into the expression

Given \( f(x) = x^{2} - 10 \), substitute \( x = 4+h \) and \( x = 4 \) into the function: \[ f(4+h) = (4+h)^{2} - 10 \] \[ f(4) = (4)^{2} - 10 \]

Step 2: Expand \( f(4+h) \)

Expand \( (4+h)^{2} \): \[ (4+h)^{2} = 16 + 8h + h^{2} \] Thus: \[ f(4+h) = 16 + 8h + h^{2} - 10 = h^{2} + 8h + 6 \]

Step 3: Simplify \( f(4) \)

Calculate \( f(4) \): \[ f(4) = 16 - 10 = 6 \]

Step 4: Compute \( f(4+h) - f(4) \)

Subtract \( f(4) \) from \( f(4+h) \): \[ f(4+h) - f(4) = (h^{2} + 8h + 6) - 6 = h^{2} + 8h \]

Step 5: Final Simplified Expression

The simplified expression is: \[ f(4+h) - f(4) = h^{2} + 8h \]

Final Answer

\(\boxed{h^{2} + 8h}\)

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