Math / Algebra

QuestionGraph of $f$
9. The graph of $f$ is shown above. Which of the following could be the equation for $f$ ? (A) $f(x)=2 \log _{4}(x)$ (B) $f(x)=-2 \log _{4}(x)$ (C) $f(x)=-2\left(\frac{1}{2}\right)^{x}$ (D) $f(x)=-2(3)^{x}$

Studdy Solution

STEP 1

1. The graph represents a function that exhibits exponential decay.
2. The function is defined for positive values of $x$ and decreases as $x$ increases.
3. The graph is in the first quadrant, indicating positive values for $f(x)$ .

STEP 2

1. Analyze the characteristics of the graph.
2. Determine the type of function that matches these characteristics.
3. Evaluate the given options to find the correct equation.

STEP 3

The graph shows a curve that starts high and decreases towards the x-axis as $x$ increases, which is characteristic of an exponential decay function.

STEP 4

Exponential decay functions generally have the form $f(x) = a \cdot b^x$ where $0 < b < 1$ .

STEP 5

Evaluate the given options:
- Option (A) $f(x) = 2 \log_4(x)$ : This is a logarithmic function, not exponential decay. - Option (B) $f(x) = -2 \log_4(x)$ : This is also a logarithmic function, not exponential decay. - Option (C) $f(x) = -2\left(\frac{1}{2}\right)^x$ : This is an exponential decay function since $\frac{1}{2} < 1$ . - Option (D) $f(x) = -2(3)^x$ : This is an exponential growth function since $3 > 1$ .

STEP 6

The correct option that represents an exponential decay function is Option (C) $f(x) = -2\left(\frac{1}{2}\right)^x$ .
The equation that could represent the graph is:
$\boxed{-2\left(\frac{1}{2}\right)^x}$

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QuestionGraph of $f$
9. The graph of $f$ is shown above. Which of the following could be the equation for $f$ ? (A) $f(x)=2 \log _{4}(x)$ (B) $f(x)=-2 \log _{4}(x)$ (C) $f(x)=-2\left(\frac{1}{2}\right)^{x}$ (D) $f(x)=-2(3)^{x}$

Studdy Solution

STEP 1

1. The graph represents a function that exhibits exponential decay.
2. The function is defined for positive values of $x$ and decreases as $x$ increases.
3. The graph is in the first quadrant, indicating positive values for $f(x)$ .

STEP 2

1. Analyze the characteristics of the graph.
2. Determine the type of function that matches these characteristics.
3. Evaluate the given options to find the correct equation.

STEP 3

The graph shows a curve that starts high and decreases towards the x-axis as $x$ increases, which is characteristic of an exponential decay function.

STEP 4

Exponential decay functions generally have the form $f(x) = a \cdot b^x$ where $0 < b < 1$ .

STEP 5

STEP 6

The correct option that represents an exponential decay function is Option (C) $f(x) = -2\left(\frac{1}{2}\right)^x$ .
The equation that could represent the graph is:
$\boxed{-2\left(\frac{1}{2}\right)^x}$

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Studdy Solution

STEP 1

1. The graph represents a function that exhibits exponential decay.2. The function is defined for positive values of x x x and decreases as x x x increases.3. The graph is in the first quadrant, indicating positive values for f(x) f(x) f(x).

STEP 2

1. Analyze the characteristics of the graph.2. Determine the type of function that matches these characteristics.3. Evaluate the given options to find the correct equation.

STEP 3

The graph shows a curve that starts high and decreases towards the x-axis as x x x increases, which is characteristic of an exponential decay function.

STEP 4

Exponential decay functions generally have the form f(x)=a⋅bx f(x) = a \cdot b^x f(x)=a⋅bx where 0<b<1 0 < b < 1 0<b<1.

STEP 5

STEP 6

The correct option that represents an exponential decay function is Option (C) f(x)=−2(12)x f(x) = -2\left(\frac{1}{2}\right)^x f(x)=−2(21​)x. The equation that could represent the graph is:−2(12)x \boxed{-2\left(\frac{1}{2}\right)^x} −2(21​)x​

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1. The graph represents a function that exhibits exponential decay.
2. The function is defined for positive values of $x$ and decreases as $x$ increases.
3. The graph is in the first quadrant, indicating positive values for $f(x)$ .

1. Analyze the characteristics of the graph.
2. Determine the type of function that matches these characteristics.
3. Evaluate the given options to find the correct equation.

The graph shows a curve that starts high and decreases towards the x-axis as $x$ increases, which is characteristic of an exponential decay function.

Exponential decay functions generally have the form $f(x) = a \cdot b^x$ where $0 < b < 1$ .

The correct option that represents an exponential decay function is Option (C) $f(x) = -2\left(\frac{1}{2}\right)^x$ .
The equation that could represent the graph is:
$\boxed{-2\left(\frac{1}{2}\right)^x}$