Question

Match the function with its value.

Answer 1

Answer:

$(f+g)(x) = x^2-3x+10$

$(f*h)(x)= 3x^3+13x^2-5x+25$

$h[f(5)]-g[h(1)]=268$

Step-by-step explanation:

1. $(f+g)(x)$ asks you to add the two functions $f(x)$ and $g(x)$, which would be $(x+5) + (x^2-4x+5)$

Simplify the like-terms and you get:

$=x^2-3x+10$

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2. $(f*h)(x)$ asks you to multiply the two functions $f(x)$ and $h(x)$, which would be $(x+5)(3x^2-2x+5)$

In order to multiply the functions we use the FOIL method (for polynomials):

$(x)(3x^2)+(x)(-2x)+(x)(5)+(5)(3x^2)+(5)(-2x)+(5)(5)$

Multiply each term and you get:

$3x^3-2x^2+5x+15x^2-10x+25$

Simplify the like-terms and you get:

$3x^3+13x^2-5x+25$

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3. $h[f(5)]-g[h(1)]$ asks for your ability to understand what composite functions are. In order to solve this, we will follow some careful steps and solve them separately.

$h[f(x)]= h[x+5]$

--> $h[f(5)]= h[5+5] = h[10]$

--> $h(10) = 3(10)^2-2(10)+5=285$

$g[h(x)] = g[3x^2-2x+5]$

--> $g[h(1)] = g[3(1)^2-2(1)+5] = g[6]$

--> $g(6) = (6)^2-4(6)+5=17$

Now that we have both values, we can just subtract them:

$285 - 17 = 268$