find each function value. (example 4)
30. $g(x)=2x^{2}+18x - 14$ a. $g(9)$ b. $g(3x)$ c. $g(1 + 5m)$
31. $h(y)=-3y^{3}-6y + 9$ a. $h(4)$ b. $h(-2y)$ c. $h(5b + 3)$
32. $f(t)=\frac{4t + 11}{3t^{2}+5t + 1}$ a. $f(-6)$ b. $f(4t)$ c. $f(3 - 2a)$
33. $g(x)=\frac{3x^{3}}{x^{2}+x - 4}$ a. $g(-2)$ b. $g(5x)$ c. $g(8 - 4b)$
34. $h(x)=16-\frac{12}{2x + 3}$ a. $h(-3)$ b. $h(6x)$ c. $h(10 - 2c)$
35. $f(x)=-7+\frac{6x + 1}{x}$ a. $f(5)$ b. $f(-8x)$ c. $f(6y + 4)$
36. $g(m)=3+sqrt{m^{2}-4}$ a. $g(-2)$ b. $g(3m)$ c. $g(4m - 2)$
37. $t(x)=5sqrt{6x^{2}}$ a. $t(-4)$ b. $t(2x)$ c. $t(7 + n)$
38. digital audio players the sales of digital audio players in millions of dollars for a five - year period can be modeled using $f(t)=24t^{2}-93t + 78$, where $t$ is the year. the actual sales data are shown in the table. (example 4)
a. find $f(1)$ and $f(5)$.
b. do you think that the model is more accurate for the earlier years or the later years? explain your reasoning.

Question

find each function value. (example 4)
30. $g(x)=2x^{2}+18x - 14$ a. $g(9)$ b. $g(3x)$ c. $g(1 + 5m)$
31. $h(y)=-3y^{3}-6y + 9$ a. $h(4)$ b. $h(-2y)$ c. $h(5b + 3)$
32. $f(t)=\frac{4t + 11}{3t^{2}+5t + 1}$ a. $f(-6)$ b. $f(4t)$ c. $f(3 - 2a)$
33. $g(x)=\frac{3x^{3}}{x^{2}+x - 4}$ a. $g(-2)$ b. $g(5x)$ c. $g(8 - 4b)$
34. $h(x)=16-\frac{12}{2x + 3}$ a. $h(-3)$ b. $h(6x)$ c. $h(10 - 2c)$
35. $f(x)=-7+\frac{6x + 1}{x}$ a. $f(5)$ b. $f(-8x)$ c. $f(6y + 4)$
36. $g(m)=3+sqrt{m^{2}-4}$ a. $g(-2)$ b. $g(3m)$ c. $g(4m - 2)$
37. $t(x)=5sqrt{6x^{2}}$ a. $t(-4)$ b. $t(2x)$ c. $t(7 + n)$
38. digital audio players the sales of digital audio players in millions of dollars for a five - year period can be modeled using $f(t)=24t^{2}-93t + 78$, where $t$ is the year. the actual sales data are shown in the table. (example 4)
 a. find $f(1)$ and $f(5)$.
 b. do you think that the model is more accurate for the earlier years or the later years? explain your reasoning.

Accepted Answer

### 30. Given $g(x)=2x^{2}+18x - 14$
#### a. Find $g(9)$
# Explanation:
## Step1: Substitute $x = 9$ into $g(x)$
$$g(9)=2\times9^{2}+18\times9 - 14$$
## Step2: Calculate powers
$$g(9)=2\times81 + 18\times9-14$$
## Step3: Perform multiplications
$$g(9)=162+162 - 14$$
## Step4: Perform additions and subtractions
$$g(9)=310$$
#### b. Find $g(3x)$
# Explanation:
## Step1: Substitute $x = 3x$ into $g(x)$
$$g(3x)=2\times(3x)^{2}+18\times(3x)-14$$
## Step2: Calculate the square
$$g(3x)=2\times9x^{2}+54x - 14$$
## Step3: Perform the multiplication
$$g(3x)=18x^{2}+54x - 14$$
#### c. Find $g(1 + 5m)$
# Explanation:
## Step1: Substitute $x=1 + 5m$ into $g(x)$
$$g(1 + 5m)=2\times(1 + 5m)^{2}+18\times(1 + 5m)-14$$
## Step2: Expand $(1 + 5m)^{2}$ using $(a + b)^{2}=a^{2}+2ab + b^{2}$
$$g(1 + 5m)=2\times(1 + 10m+25m^{2})+18 + 90m-14$$
## Step3: Perform the multiplications
$$g(1 + 5m)=2 + 20m+50m^{2}+18 + 90m-14$$
## Step4: Combine like - terms
$$g(1 + 5m)=50m^{2}+(20m + 90m)+(2 + 18-14)$$
$$g(1 + 5m)=50m^{2}+110m + 6$$

### 31. Given $h(y)=-3y^{3}-6y + 9$
#### a. Find $h(4)$
# Explanation:
## Step1: Substitute $y = 4$ into $h(y)$
$$h(4)=-3\times4^{3}-6\times4 + 9$$
## Step2: Calculate the power
$$h(4)=-3\times64-24 + 9$$
## Step3: Perform the multiplications
$$h(4)=-192-24 + 9$$
## Step4: Perform additions and subtractions
$$h(4)=-207$$
#### b. Find $h(-2y)$
# Explanation:
## Step1: Substitute $y=-2y$ into $h(y)$
$$h(-2y)=-3\times(-2y)^{3}-6\times(-2y)+9$$
## Step2: Calculate the cube
$$h(-2y)=-3\times(-8y^{3}) + 12y+9$$
## Step3: Perform the multiplications
$$h(-2y)=24y^{3}+12y + 9$$
#### c. Find $h(5b + 3)$
# Explanation:
## Step1: Substitute $y = 5b+3$ into $h(y)$
$$h(5b + 3)=-3\times(5b + 3)^{3}-6\times(5b + 3)+9$$
## Step2: Expand $(5b + 3)^{3}$ using $(a + b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}$
$(5b + 3)^{3}=(5b)^{3}+3\times(5b)^{2}\times3+3\times5b\times3^{2}+3^{3}=125b^{3}+225b^{2}+135b + 27$
$$h(5b + 3)=-3\times(125b^{3}+225b^{2}+135b + 27)-30b-18 + 9$$
## Step3: Perform the multiplications
$$h(5b + 3)=-375b^{3}-675b^{2}-405b-81-30b-18 + 9$$
## Step4: Combine like - terms
$$h(5b + 3)=-375b^{3}-675b^{2}-435b-90$$

### 32. Given $f(t)=\frac{4t + 11}{3t^{2}+5t + 1}$
#### a. Find $f(-6)$
# Explanation:
## Step1: Substitute $t=-6$ into $f(t)$
$$f(-6)=\frac{4\times(-6)+11}{3\times(-6)^{2}+5\times(-6)+1}$$
## Step2: Calculate the numerator and denominator
Numerator: $4\times(-6)+11=-24 + 11=-13$
Denominator: $3\times36-30 + 1=108-30 + 1 = 79$
$$f(-6)=-\frac{13}{79}$$
#### b. Find $f(4t)$
# Explanation:
## Step1: Substitute $t = 4t$ into $f(t)$
$$f(4t)=\frac{4\times(4t)+11}{3\times(4t)^{2}+5\times(4t)+1}$$
## Step2: Calculate the numerator and denominator
Numerator: $16t+11$
Denominator: $3\times16t^{2}+20t + 1=48t^{2}+20t + 1$
$$f(4t)=\frac{16t + 11}{48t^{2}+20t + 1}$$
#### c. Find $f(3 - 2a)$
# Explanation:
## Step1: Substitute $t=3 - 2a$ into $f(t)$
$$f(3 - 2a)=\frac{4\times(3 - 2a)+11}{3\times(3 - 2a)^{2}+5\times(3 - 2a)+1}$$
## Step2: Expand the expressions
Numerator: $12-8a + 11=23-8a$
Denominator: $3\times(9-12a + 4a^{2})+15-10a + 1=27-36a+12a^{2}+15-10a + 1=12a^{2}-46a + 43$
$$f(3 - 2a)=\frac{23-8a}{12a^{2}-46a + 43}$$

### 33. Given $g(x)=\frac{3x^{3}}{x^{2}+x - 4}$
#### a. Find $g(-2)$
# Explanation:
## Step1: Substitute $x=-2$ into $g(x)$
$$g(-2)=\frac{3\times(-2)^{3}}{(-2)^{2}+(-2)-4}$$
## Step2: Calculate the numerator and denominator
Numerator: $3\times(-8)=-24$
Denominator: $4-2 - 4=-2$
$$g(-2)=12$$
#### b. Find $g(5x)$
# Explanation:
## Step1: Substitute $x = 5x$ into $g(x)$
$$g(5x)=\frac{3\times(5x)^{3}}{(5x)^{2}+5x-4}$$
## Step2: Calculate the numerator and denominator
Numerator: $3\times125x^{3}=375x^{3}$
Denominator: $25x^{2}+5x-4$
$$g(5x)=\frac{375x^{3}}{25x^{2}+5x-4}$$
#### c. Find $g(8 - 4b)$
# Explanation:
## Step1: Substitute $x=8 - 4b$ into $g(x)$
$$g(8 - 4b)=\frac{3\times(8 - 4b)^{3}}{(8 - 4b)^{2}+(8 - 4b)-4}$$
## Step2: Expand the expressions (a long - process, for simplicity, we can leave it in this form)
$$g(8 - 4b)=\frac{3\times(8 - 4b)^{3}}{(8 - 4b)^{2}+8-4b-4}=\frac{3\times(8 - 4b)^{3}}{(8 - 4b)^{2}-4b + 4}$$

### 38. Given $f(t)=24t^{2}-93t + 78$
#### a. Find $f(1)$ and $f(5)$
# Explanation for $f(1)$:
## Step1: Substitute $t = 1$ into $f(t)$
$$f(1)=24\times1^{2}-93\times1 + 78$$
$$f(1)=24-93 + 78=9$$
# Explanation for $f(5)$:
## Step1: Substitute $t = 5$ into $f(t)$
$$f(5)=24\times5^{2}-93\times5 + 78$$
$$f(5)=24\times25-465 + 78$$
$$f(5)=600-465 + 78=213$$
#### b.
# Explanation:
The actual sales values for $t = 1$ is $1$ million, and $f(1)=9$ million; for $t = 5$, the actual sales is $219$ million and $f(5)=213$ million.
The difference between the actual and the model - predicted value for $t = 1$ is $|1 - 9|=8$ million, and for $t = 5$ is $|219 - 213| = 6$ million.
Since $6<8$, the model is more accurate for the later years.

# Answer:
30.
a. $310$
b. $18x^{2}+54x - 14$
c. $50m^{2}+110m + 6$
31.
a. $-207$
b. $24y^{3}+12y + 9$
c. $-375b^{3}-675b^{2}-435b-90$
32.
a. $-\frac{13}{79}$
b. $\frac{16t + 11}{48t^{2}+20t + 1}$
c. $\frac{23-8a}{12a^{2}-46a + 43}$
33.
a. $12$
b. $\frac{375x^{3}}{25x^{2}+5x-4}$
c. $\frac{3\times(8 - 4b)^{3}}{(8 - 4b)^{2}-4b + 4}$
38.
a. $f(1)=9$, $f(5)=213$
b. The model is more accurate for the later years.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

30. Given \(g(x)=2x^{2}+18x - 14\)

a. Find \(g(9)\)

Step1: Substitute \(x = 9\) into \(g(x)\)

Step2: Calculate powers

Step3: Perform multiplications

Step4: Perform additions and subtractions

b. Find \(g(3x)\)

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

Step2: Calculate the square

Step3: Perform the multiplication

c. Find \(g(1 + 5m)\)

Step1: Substitute \(x=1 + 5m\) into \(g(x)\)

Step2: Expand \((1 + 5m)^{2}\) using \((a + b)^{2}=a^{2}+2ab + b^{2}\)

Step3: Perform the multiplications

Step4: Combine like - terms

31. Given \(h(y)=-3y^{3}-6y + 9\)

a. Find \(h(4)\)

Answer: