Question

category 22 – practice question
after the equations in the system are determined, the desmos
question 6
1* easy
the function f is defined by f(x) = x⁴ − x³ + 18.
what is the value of f(x) when x = 3?
2* easy
the function f is defined by f(x) = x⁴ − 3x² − 4.
what is the value of f(−2) − f(1)?
a) −6
b) −2
c) 2
d) 6
3* medium
f(x) = 4x⁴ − 57
the function f is defined as shown above, and
3f(b) = 21. what is the positive value of b?
a) 0
b) 2
c) 4
d) 7

Explanation:

1st Question

Step1: Substitute \( x = 3 \) into \( f(x) \)

Substitute \( x = 3 \) into \( f(x)=x^{4}-x^{3}+18 \), we get \( f(3)=3^{4}-3^{3}+18 \).

Step2: Calculate each term

Calculate \( 3^{4}=81 \), \( 3^{3}=27 \). Then \( f(3)=81 - 27+18 \).

Step3: Simplify the expression

\( 81-27 = 54 \), \( 54 + 18=72 \).

Step1: Find \( f(-2) \)

Substitute \( x=-2 \) into \( f(x)=x^{4}-3x^{2}-4 \), we have \( f(-2)=(-2)^{4}-3\times(-2)^{2}-4 \). Calculate \( (-2)^{4}=16 \), \( (-2)^{2}=4 \), so \( f(-2)=16-3\times4 - 4=16 - 12-4 = 0 \).

Step2: Find \( f(1) \)

Substitute \( x = 1 \) into \( f(x)=x^{4}-3x^{2}-4 \), we get \( f(1)=1^{4}-3\times1^{2}-4=1 - 3-4=-6 \).

Step3: Calculate \( f(-2)-f(1) \)

\( f(-2)-f(1)=0-(-6)=6 \).

Step1: Solve for \( f(b) \)

Given \( 3f(b)=21 \), divide both sides by 3, we get \( f(b)=\frac{21}{3}=7 \).

Step2: Substitute \( f(b) \) into the function

Since \( f(x)=4x^{4}-57 \), then \( 4b^{4}-57 = 7 \).

Step3: Solve for \( b^{4} \)

Add 57 to both sides: \( 4b^{4}=7 + 57=64 \). Divide both sides by 4: \( b^{4}=\frac{64}{4}=16 \).

Step4: Solve for \( b \)

Take the fourth - root of both sides. Since we want the positive value, \( b=\sqrt[4]{16}=2 \) (because \( 2^{4}=16 \)).

Answer:

\( 72 \)

2nd Question