If $a$ and $h$ are real numbers, find the following values for the given function.

$f(x)=x^2-x+24$

(a) $f(a)$

(b) $f(-a)$

(c) $-f(a)$

(d) $f(a+h)$

(e) $f(a)+f(h)$

(f) $\frac{f(a+h)-f(a)}{h}$, if $h \neq 0$

Evaluate f ( a ) by substituting a into f ( x ) , resulting in f ( a ) = a 2 − a + 24 .
Evaluate f ( − a ) by substituting − a into f ( x ) , resulting in f ( − a ) = a 2 + a + 24 .
Evaluate − f ( a ) by multiplying f ( a ) by − 1 , resulting in − f ( a ) = − a 2 + a − 24 .
Evaluate f ( a + h ) by substituting a + h into f ( x ) , resulting in f ( a + h ) = a 2 + 2 ah + h 2 − a − h + 24 .
Evaluate f ( a ) + f ( h ) by adding f ( a ) and f ( h ) , resulting in f ( a ) + f ( h ) = a 2 − a + h 2 − h + 48 .
Evaluate h f ( a + h ) − f ( a ) ​ by simplifying the expression, resulting in h f ( a + h ) − f ( a ) ​ = 2 a + h − 1 .

f ( a ) = a 2 − a + 24 , f ( − a ) = a 2 + a + 24 , − f ( a ) = − a 2 + a − 24 , f ( a + h ) = a 2 + 2 ah + h 2 − a − h + 24 , f ( a ) + f ( h ) = a 2 − a + h 2 − h + 48 , h f ( a + h ) − f ( a ) ​ = 2 a + h − 1 ​
Explanation

Understanding the problem We are given the function f ( x ) = x 2 − x + 24 , and we need to find expressions for f ( a ) , f ( − a ) , − f ( a ) , f ( a + h ) , f ( a ) + f ( h ) , and h f ( a + h ) − f ( a ) ​ . We will substitute the given expressions into the function and simplify.

Finding f(a) (a) To find f ( a ) , we replace x with a in the expression for f ( x ) : f ( a ) = a 2 − a + 24

Finding f(-a) (b) To find f ( − a ) , we replace x with − a in the expression for f ( x ) : f ( − a ) = ( − a ) 2 − ( − a ) + 24 = a 2 + a + 24

Finding -f(a) (c) To find − f ( a ) , we multiply the expression for f ( a ) by − 1 : − f ( a ) = − ( a 2 − a + 24 ) = − a 2 + a − 24

Finding f(a+h) (d) To find f ( a + h ) , we replace x with a + h in the expression for f ( x ) : f ( a + h ) = ( a + h ) 2 − ( a + h ) + 24 = a 2 + 2 ah + h 2 − a − h + 24

Finding f(a) + f(h) (e) To find f ( a ) + f ( h ) , we first find f ( h ) by replacing x with h in the expression for f ( x ) : f ( h ) = h 2 − h + 24 Then we add f ( a ) and f ( h ) : f ( a ) + f ( h ) = ( a 2 − a + 24 ) + ( h 2 − h + 24 ) = a 2 − a + h 2 − h + 48

Finding (f(a+h) - f(a))/h (f) To find h f ( a + h ) − f ( a ) ​ , we first find f ( a + h ) − f ( a ) : f ( a + h ) − f ( a ) = ( a 2 + 2 ah + h 2 − a − h + 24 ) − ( a 2 − a + 24 ) = 2 ah + h 2 − h Then we divide by h : h f ( a + h ) − f ( a ) ​ = h 2 ah + h 2 − h ​ = 2 a + h − 1

Final Answer Therefore, the values are: (a) f ( a ) = a 2 − a + 24 (b) f ( − a ) = a 2 + a + 24 (c) − f ( a ) = − a 2 + a − 24 (d) f ( a + h ) = a 2 + 2 ah + h 2 − a − h + 24 (e) f ( a ) + f ( h ) = a 2 − a + h 2 − h + 48 (f) h f ( a + h ) − f ( a ) ​ = 2 a + h − 1


Examples
Understanding function transformations is crucial in many fields. For example, in physics, if f ( x ) represents the position of an object at time x , then f ( a + h ) could represent the position of the object at a future time a + h . The expression h f ( a + h ) − f ( a ) ​ gives the average velocity of the object over the time interval [ a , a + h ] . By manipulating and analyzing these expressions, physicists can gain insights into the motion of objects. Similarly, in economics, functions can model cost, revenue, or profit, and understanding transformations helps in analyzing changes in these quantities.

Answered by GinnyAnswer | 2025-07-07

The evaluations of the function are: f ( a ) = a 2 − a + 24 , f ( − a ) = a 2 + a + 24 , − f ( a ) = − a 2 + a − 24 , f ( a + h ) = a 2 + 2 ah + h 2 − a − h + 24 , f ( a ) + f ( h ) = a 2 − a + h 2 − h + 48 , and h f ( a + h ) − f ( a ) ​ = 2 a + h − 1 .
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Answered by Anonymous | 2025-08-23