Question

skill #5: function notation
find ( f(-4) ) for each function.

1. ( f(x) = 2x^2 + 3x - 7 )
2. ( f(x) = \frac{2}{x} )
3. ( f(t) = t^3 - \frac{t}{2} )

solve for the missing variable.

1. what value of ( x ) makes ( f(x) = 8 ) if ( f(x) = 3x + 12 )?
2. what value(s) of ( t ) make ( f(t) = 2 ), if ( f(t) = t^2 - 7 ).

evaluate the function.

1. if ( f(x) = 2x + 3 ), what is the function for ( f(x + 2) )?
2. if ( f(x) = 2x + 3 ), what is the function for ( f(2x) )?

Explanation:

31. Step1: Substitute $x=-4$

$f(-4)=2(-4)^2+3(-4)-7$

31. Step2: Calculate each term

$2(16) + (-12) - 7 = 32 - 12 - 7$

31. Step3: Simplify the expression

$32-12-7=13$

32. Step1: Substitute $x=-4$

$f(-4)=\frac{2}{-4}$

32. Step2: Simplify the fraction

$\frac{2}{-4}=-\frac{1}{2}$

33. Step1: Substitute $t=-4$

$f(-4)=(-4)^3-\frac{-4}{2}$

33. Step2: Calculate each term

$-64 - (-2) = -64 + 2$

33. Step3: Simplify the expression

$-64+2=-62$

34. Step1: Set $f(x)=8$, substitute function

$8=3x+12$

34. Step2: Isolate $3x$ term

$3x=8-12$

34. Step3: Solve for $x$

$x=\frac{-4}{3}$

35. Step1: Set $f(t)=2$, substitute function

$2=t^2-7$

35. Step2: Isolate $t^2$ term

$t^2=2+7=9$

35. Step3: Solve for $t$

$t=\pm\sqrt{9}=\pm3$

36. Step1: Substitute $x+2$ into $f(x)$

$f(x+2)=2(x+2)+3$

36. Step2: Expand and simplify

$2x+4+3=2x+7$

37. Step1: Substitute $2x$ into $f(x)$

$f(2x)=2(2x)+3$

37. Step2: Simplify the expression

$4x+3$

Answer:

1. $13$
2. $-\frac{1}{2}$
3. $-62$
4. $x=-\frac{4}{3}$
5. $t=3$ and $t=-3$
6. $f(x+2)=2x+7$
7. $f(2x)=4x+3$