Multiple choice math questions.

Steps to Solve

1. Question 3: Evaluate the functions at x=3

We need to plug in $x=3$ into each function and find the largest value. A. $f(3) = 2 \cdot 4^3 = 2 \cdot 64 = 128$ B. $f(3) = 4 \cdot 2^3 = 4 \cdot 8 = 32$ C. $f(3) = 4 \cdot 3^2 = 4 \cdot 9 = 36$ D. $f(3) = 2 \cdot 3^4 = 2 \cdot 81 = 162$

2. Question 3: Determine the greatest value

Comparing the values from step 1, the greatest value is 162. Therefore, option D has the greatest value at $x=3$.

3. Question 4: Identify functions with x-intercepts

An x-intercept occurs where $y=0$. We need to find which function cannot equal 0. F. $y = 2x + 3$. Setting $y=0$, we get $2x+3=0$, so $x = -3/2$. Thus, there is an x-intercept. G. $y = 2x^3$. Setting $y=0$, we get $2x^3=0$, so $x=0$. Thus, there is an x-intercept. H. $y = 3x^2$. Setting $y=0$, we get $3x^2=0$, so $x=0$. Thus, there is an x-intercept. I. $y = 3 \cdot 2^x$. The value of $2^x$ will always be greater than 0 for any real number $x$. Multiplying by 3 does not change this. Therefore, $y$ can never be 0, and there is no x-intercept.

4. Question 5: Evaluate each statement about $y=2^x$

We need to determine which statement is NOT true. A. The function is an exponential function. This is true based on the form of the equation. B. The function has a domain of all real numbers. This is also true; we can plug in any real number for $x$. C. As the value of $x$ gets very large, the value of $y$ gets close to zero. This is false. As $x$ gets very large, $y=2^x$ also gets very large. As $x$ gets very small (large negative values), $y$ gets close to zero. D. As the value of $x$ increases by one, the value of $y$ doubles. This is true. For example, $2^{x+1} = 2^x \cdot 2^1 = 2 \cdot 2^x$.

5. Question 6: Test each ordered pair

We need to find which ordered pair satisfies the equation $y = 3 \cdot 4^x$. F. $(0,0)$: $3 \cdot 4^0 = 3 \cdot 1 = 3 \neq 0$. G. $(1,81)$: $3 \cdot 4^1 = 3 \cdot 4 = 12 \neq 81$. H. $(2,24)$: $3 \cdot 4^2 = 3 \cdot 16 = 48 \neq 24$. I. $(2,48)$: $3 \cdot 4^2 = 3 \cdot 16 = 48$. This ordered pair works.