Value of a Function

Value of a function. $f(a)$ is called the value of the function $f(x)$ at $x=a$ — simply the result of substituting $x=a$ into the formula for $f$.

Ex. 1 (Evaluation): Evaluate $f(x)=2x^2-3x+4$ at $x=7$ and $x=-2t$.

Solution: $f(7)=2(7)^2-3(7)+4=2(49)-21+4=98-21+4=81$.

$f(-2t)=2(-2t)^2-3(-2t)+4=2(4t^2)+6t+4=8t^2+6t+4$.

Ex. 2 (Reading values off a graph): Using the graph of $y=g(x)$ (Fig. 6.14), find $g(-4)$ and $g(3)$.

Solution: From the graph, when $x=-4$, $y=0$, so $g(-4)=0$. When $x=3$, $y=-5$, so $g(3)=-5$.

Ex. 3 (Solving 'backwards' from an output, algebraically): If $t(m)=3m^2-m$ and $t(m)=4$, find $m$.

Solution: $3m^2-m=4 \implies 3m^2-m-4=0$. Factor by splitting the middle term: $3m^2-4m+3m-4=0 \implies m(3m-4)+1(3m-4)=0 \implies (3m-4)(m+1)=0$. So $m=\dfrac43$ or $m=-1$.

Ex. 4 (Solving 'backwards' from a graph): From the graph below (Fig. 6.15), find $x$ for which $f(x)=4$.

Solution: Solving $f(x)=4$ means finding where the graph meets the horizontal line $y=4$. Reading the intersection points off the graph (Fig. 6.16), the line $y=4$ meets the curve at $x=-1$ and $x=3$. So $x=-1$ and $x=3$. …