Pergunta

A function g is given by g(x)=x^2-9 Find g(-6),g(0),g(6),g(a+h) and (g(x+h)-g(x))/(h) g(-6)=square (Simplify your answer.)
Solução

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AfonsoVeterano · Tutor por 11 anos
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To find \( g(-6) \), we need to substitute \( x = -6 \) into the function \( g(x) = x^2 - 9 \).<br /><br />\[ g(-6) = (-6)^2 - 9 \]<br /><br />Calculate \( (-6)^2 \):<br /><br />\[ (-6)^2 = 36 \]<br /><br />Now, subtract 9:<br /><br />\[ 36 - 9 = 27 \]<br /><br />So, \( g(-6) = 27 \).<br /><br />Next, let's find \( g(0) \):<br /><br />\[ g(0) = 0^2 - 9 \]<br /><br />Since \( 0^2 = 0 \):<br /><br />\[ 0 - 9 = -9 \]<br /><br />So, \( g(0) = -9 \).<br /><br />Now, let's find \( g(6) \):<br /><br />\[ g(6) = 6^2 - 9 \]<br /><br />Calculate \( 6^2 \):<br /><br />\[ 6^2 = 36 \]<br /><br />Now, subtract 9:<br /><br />\[ 36 - 9 = 27 \]<br /><br />So, \( g(6) = 27 \).<br /><br />Next, let's find \( g(a+h) \):<br /><br />\[ g(a+h) = (a+h)^2 - 9 \]<br /><br />Expand \( (a+h)^2 \):<br /><br />\[ (a+h)^2 = a^2 + 2ah + h^2 \]<br /><br />So,<br /><br />\[ g(a+h) = a^2 + 2ah + h^2 - 9 \]<br /><br />Finally, let's find \( \frac{g(x+h) - g(x)}{h} \):<br /><br />First, find \( g(x+h) \):<br /><br />\[ g(x+h) = (x+h)^2 - 9 \]<br /><br />Expand \( (x+h)^2 \):<br /><br />\[ (x+h)^2 = x^2 + 2xh + h^2 \]<br /><br />So,<br /><br />\[ g(x+h) = x^2 + 2xh + h^2 - 9 \]<br /><br />Now, find \( g(x) \):<br /><br />\[ g(x) = x^2 - 9 \]<br /><br />Subtract \( g(x) \) from \( g(x+h) \):<br /><br />\[ g(x+h) - g(x) = (x^2 + 2xh + h^2 - 9) - (x^2 - 9) \]<br /><br />Simplify the expression:<br /><br />\[ g(x+h) - g(x) = x^2 + 2xh + h^2 - 9 - x^2 + 9 \]<br /><br />\[ g(x+h) - g(x) = 2xh + h^2 \]<br /><br />Now, divide by \( h \):<br /><br />\[ \frac{g(x+h) - g(x)}{h} = \frac{2xh + h^2}{h} \]<br /><br />Factor out \( h \) in the numerator:<br /><br />\[ \frac{g(x+h) - g(x)}{h} = \frac{h(2x + h)}{h} \]<br /><br />Cancel \( h \) in the numerator and denominator:<br /><br />\[ \frac{g(x+h) - g(x)}{h} = 2x + h \]<br /><br />So, the final answers are:<br /><br />\[ g(-6) = 27 \]<br />\[ g(0) = -9 \]<br />\[ g(6) = 27 \]<br />\[ g(a+h) = a^2 + 2ah + h^2 - 9 \]<br />\[ \frac{g(x+h) - g(x)}{h} = 2x + h \]
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