The function $\boldsymbol{f}$ is defined by $\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{a}^{\boldsymbol{x}}+\boldsymbol{b}$, where $\boldsymbol{a}$ and $\boldsymbol{b}$ are constants. In the $\boldsymbol{x y}$-plane, the graph of $\boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x})$ has an $\boldsymbol{x}$-intercept at $(\mathbf{2}, \mathbf{0})$ and a $\boldsymbol{y}$-intercept at $(\mathbf{0},-\mathbf{3 2 3})$. What is the value of $\boldsymbol{b}$ ?
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Answer

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Step 1
: Recall that the x-intercept of a function is the point where the graph crosses the x-axis.

f(2)=a2+b=0(1)f(2) = a^{2} + b = 0 \quad (1)
At this point, the y-coordinate is 0. Since the given function has an x-intercept at (2, 0), we can find the value of a by substituting x = 2 and y = 0 into the function.

Step 2
: We also know that the y-intercept of a function is the point where the graph crosses the y-axis.

f(0)=a0+b=323(2)f(0) = a^{0} + b = -323 \quad (2)
At this point, the x-coordinate is 0. The given function has a y-intercept at (0, - 323). Since the y-intercept is (- 323, 0) in terms of the function, we can find the value of b by substituting x = 0 and y = - 323 into the function.

Final Answer

The value of b is - 324.